The trio join Denis Shapovalov, Felix Auger-Aliassime and Diego Schwartzman for the team event which runs from September 24-26 at TD Garden in Boston.
Laver Cup newcomer Opelka rose to a career-high world number 23 ranking en route to his first ATP Masters 1000 final in Toronto and defeated world number three Stefanos Tsitsipas, who will play for Bjorn Borg's Team Europe.
Isner, who has featured for Team World since the inaugural event in 2017, reached the semi-finals in Toronto and claimed his 16th ATP Tour title in Atlanta at the start of August.
He described the Laver Cup as "a highlight of my year", adding: "To be on a team with guys we're normally competing against is so different and so much fun. We come together so well as a group, the chemistry is awesome and it's such a great environment to be part of."
Australian firebrand Kyrgios is a striking inclusion in Team World's roster, while Team Europe will be without their big three: Roger Federer, Rafael Nadal and Novak Djokovic.
Federer and Dominic Thiem were expected to take part in this year's event, though both were forced to withdraw with injuries.
However, Borg's men still boast six of the world's top 11. World number two Daniil Medvedev leads the line-up, with Tsitsipas and Tokyo Olympics gold medallist Alexander Zverev for company.
Casper Ruud, who collected a 14th win in his last 15 completed matches on tour when he beat Opelka on Wednesday, will feature, while Andrey Rublev and Wimbledon finalist Matteo Berrettini complete the six-man team.
Team Europe have landed the title in each of the three editions of the tournament so far, with Prague, Chicago and Geneva having served as hosts.
Europe swept Saturday's four matches in Boston to stand on the cusp of a fourth consecutive Laver Cup triumph – the defending champions lead 11-1 and require just two more points to clinch the title.
Stefanos Tsitsipas blitzed Team World's Kyrgios 6-3 6-4 at TD Garden, where Olympic Games gold medallist Alexander Zverev beat John Isner 7-6 (7-5) 6-7 (6-8) 10-5 before US Open champion Daniil Medvedev made light work of Denis Shapovalov 6-4 6-0.
Team Europe secured their fourth win of the day in the doubles – Tsitsipas and Andrey Rublev teaming up to defeat Isner and Kyrgios 6-7 (8-10) 6-3 10-4.
After Kyrgios' straight-sets loss to Greece's Tsitsipas, the 26-year-old Australian star casted doubt over his tennis future.
"This is my probably my last Laver Cup," former world number 13 Kyrgios – an Australian Open and Wimbledon quarter-finalist – told reporters post-match. "I don't know how much longer I will be in tennis.
"This is my last event of the year. I will get my body right ahead of the Australian Open.
"My mum is not doing too well with her health. I'd like to go back and see her."
"As long as I'm on the court, I will try and give my best, but I'm not going to lie and say that I'm going to plan to play four or five more years on tour," Kyrgios said. "That's just not me."
Playing for the first time since earning his maiden grand slam trophy at the expense of record-chasing Novak Djokovic at Flushing Meadows, world number two Medvedev suffered no letdown against Shapovalov.
"I played unbelievably, especially [in] the second set," Russia's Medvedev said in his on-court interview. "I didn't know what to expect because after the US Open, I didn't play for a week and a half. Came here, practised as much as I could the past three days, so I didn't hit [that] many balls, but was surprisingly feeling well.
"I wanted to show that also today. [The] first [set] was not easy, the ball was not going as fast as I wanted [and] he was playing really good. And then I just couldn't miss a ball anymore. I'm really happy about [that]."
Isner has failed to recover from the wrist injury that forced him to withdraw from the US Open ahead of his second-round match against Holger Rune.
Fellow American Tiafoe knocked out Rafael Nadal en route to reaching the semi-finals at Flushing Meadows, where he was beaten by tournament winner Carlos Alcaraz.
The 24-year-old, now at a career high of 19th in the ATP rankings, will take Isner's place at the O2 Arena as Team World aim to end Team Europe's run of four triumphs in a row.
Tiafoe previously took part in the tournament in 2017 and 2018, but has missed the past two editions.
Team World captain John McEnroe said: "I am thrilled to have Frances on the team.
"He's raised his game to a whole new level and has shown he can compete and win against the best players on Tour. He brings a positive energy that should help the entire team."
Tiafoe joins Felix Auger-Aliassime, Taylor Fritz, Diego Schwartzman, Alex de Minaur and Jack Sock on McEnroe's six-man team.
Team Europe, led by Bjorn Borg, is made up of Nadal, Novak Djokovic, Stefanos Tsitsipas, Casper Ruud, Roger Federer and Andy Murray.
The tournament is contested on an indoor hard court and runs for three days from September 23.
Top seed Medvedev, who won just three games against Nadal in the final in 2019, the last time the tournament was played, prevailed 4-6 6-3 6-4 in his opener against Alexander Bublik.
The man Nadal beat to win the title the previous year, Tsitsipas, failed to convert on five match points in an epic second-set tiebreak but recovered to down Ugo Humbert 6-3 6-7 (13-15) 6-1.
Nadal pulled out ahead of a scheduled match against Lloyd Harris, who beat him last week in Washington as the 20-time grand slam winner struggled with a foot injury.
Countryman Feliciano Lopez replaces Nadal in the draw and will face Harris on Wednesday.
Elsewhere Tuesday, sixth seed Casper Ruud needed more than two hours to put away Marin Cilic 6-3 3-6 6-3, while 14th seed Grigor Dimitrov suffered a quick 6-3 6-4 exit against big-serving American Reilly Opelka.
A pair of unseeded veterans advanced, with 2016 finalist Kei Nishikori a 6-4 6-7 (5-7) 6-2 winner over Miomir Kecmanovic and John Isner defeating Alejandro Davidovich Fokini 6-4 6-1 in just over an hour.
Two days after appearing in his first ATP Tour final at the Citi Open, Mackenzie McDonald fell 6-3 6-4 to Benoit Paire in his Toronto opener.
In other first-round matches, Karen Khachanov beat Cameron Norrie 6-4 5-7 6-4, Frances Tiafoe downed Yoshihito Nishioka 6-4 6-3, Dusan Lajovic handled Emil Ruusuvuori 3-6 6-3 6-3 and Nikoloz Basilashvili defeated Jenson Brooksby 2-6 6-0 6-4.
Medvedev looked in all sorts of trouble against the Australian, but the world number two and top seed survived at the ATP 1000 tournament on Sunday.
Canadian sensation Felix Auger-Aliassime was sent packing by former champion John Isner in Miami.
MEDVEDEV ADVANCES… SOMEHOW
Australian Open and US Open runner-up Medvedev could barely walk during his gruelling three-set win against Popyrin.
Medvedev let three match points slip away in the second set, having led 5-2, before cramp set in.
In humid conditions, Medvedev was forced to rely on his serve as he limped around the court.
"Winning a Grand Slam final in straight sets doesn't feel the way I felt after the match point today," he said. "A match to remember for sure."
"I felt like my legs were not following me anymore," Medvedev said. "The only thing I was thinking about is not to fall down, because if you fall down, I don't think I would be able to get up. There were a few moments I just wanted to lay down and say, 'OK, it's over.' That's the thing I couldn't accept myself to do."
Awaiting Medvedev is Frances Tiafoe after he outlasted 16th seed Dusan Lajovic 1-6 7-5 6-3.
AUGER-ALIASSIME TOPPLED
A new year, but same result for 11th seed Auger-Aliassime, who was beaten 7-6 (7-5) 7-6 (7-5) by Isner.
Isner defeated Auger-Aliassime en route to the championship match in Miami two years ago and the big-serving American repeated the feat.
The only former Miami champion in this year's field, Isner remains unbeaten in tie-breaks at Hard Rock Stadium, winning all 11 since the tournament relocated in 2019.
"It was a big test for me. There's probably not many places in the world where I like my chances against Felix," Isner said. "I'm kind of lucky to play him here in Miami, where I like it a lot. If I were to play him on clay somewhere in Europe, it'd probably be a different story."
Next up is seventh seed Roberto Bautista-Agut, who beat Jan-Lennard Struff 4-6 6-3 6-2.
Elsewhere, Jannik Sinner, Taylor Fritz, Emil Ruusuvuori and Alexander Bublik also advanced.
Top seed and world number two Medvedev appeared headed for another defeat to Hurkacz before fighting back to prevail 2-6 7-6 (8-6) 7-6 (7-5) and reach the semi-finals.
The Pole won a five-setter at Wimbledon in the pair's only previous meeting before going on to stun Roger Federer and reach the semis, but could not repeat his feat in Toronto.
Though Medvedev failed to convert the only two break point opportunities he had, his work in the tiebreaks was enough to put him over the top as he served 23 aces to nine for Hurkacz.
The Russian now faces a semi-final matchup against John Isner, who won a clash of veterans with 11th seed Gael Monfils 7-6 (7-5) 6-4.
Earlier Friday, Tsitsipas claimed a 6-1 6-4 victory over Ruud, who entered the clash having won each of his previous 13 matches.
It was Tsitsipas' 45th win of the season, a figure unmatched by anyone else on the ATP tour, and he needed just 74 minutes to complete it.
The 23-year-old won each of the first five games of the match and only made four unforced errors in the opening set.
Although Norway's Ruud improved in the second, he failed to make the most of a rare break opportunity and that seemed to jolt Tsitsipas into life again, the Greek swiftly breaking and then serving out the match.
"It is very nice to see myself perform at this level," Tsitsipas said afterwards. "I was sticking close to the baseline and coming in, taking the ball early. It was my intention from the very beginning, and it worked perfectly.
"Playing against a guy like him, who gets every single ball back, is always very challenging. There is some sort of concentration levels you have to reach to perform to your best. These kinds of matches always teach me things that I can expose for next time."
Up next for Tsitsipas as he targets a place in the final is Reilly Opelka, the big-serving American who defeated Roberto Bautista Agut 6-3 7-6 (7-1), plundering 18 aces along the way.
World number 32 Opelka fought back from a set down in Toronto to knock out favoured third seed Tsitsipas in two hours and 32 minutes.
The 23-year-old American triumphed 6-7 (2-7) 7-6 (7-4) 6-4, booking a spot in his first ATP Masters 1000 final.
Opelka sent down 17 aces but also hit 27 groundstroke winners in the shock victory over 2021 Australian Open semi-finalist Tsitsipas who had beaten in-form Casper Ruud in the quarters.
"I played great," Opelka said in his on-court interview. “These courts suit my game, it is no coincidence that John Isner is in the other semi-final.
"Stefanos has definitely improved his returning. Come 5-5 in the first set, a lot of balls started to come back. He is a thinker and has a high tennis IQ, so it was expected, but that is what separates him from the rest of the pack."
Opelka was aided by an excellent first serve percentage of 72 per cent, winning 62 of 81 points when he made his first serve.
Both first two sets went to tiebreaks, with Tsitsipas failing to generate a break point until the third set. Opelka made the first break of the match in the seventh game of the third set and held serve twice after to seal victory.
World number two Medvedev cruised into the decider with a commanding win over John Isner in less than an hour.
Medvedev, who was runner-up in Toronto in 2019, won 6-2 6-2 over the big-serving American, with the match totaling 54 minutes.
Isner could only manage four trademark aces for the match, while the Russian had 11 and broke the American four times from seven opportunities.
The 37-year-old Frenchman has not played competitively since August last year due to injury but was eliminated by Thompson 6-3 6-1 in one hour and five minutes.
The Australian was ruthless, converting four of six break points generated, while he did not offer up any to Monfils.
Monfils was rusty, committing 17 unforced errors to Thompson's five, but the veteran hit 14 winners.
Thompson's victory earns him a second-round meeting with second seed Stefanos Tsitsipas.
Brandon Nakashima won the battle of the Americans, getting past John Isner 7-6 (9-7) 6-3 in one hour and 39 minutes. Nakashima progresses to face fifth seed Daniil Medvedev.
China's Wu Yibing produced some thrilling tennis, including an astonishing volley, in his 6-7 (6-8) 6-0 7-6 (7-3) victory over Jaume Munar.
World number 38 Diego Schwartzman won 6-1 6-2 over fellow Argentinian Federico Coria, while 47th-ranked Jiri Lehecka got past Arthur Rinderknech 7-6 (7-4) 7-5.
Australia's Jason Kubler toppled Lorenzo Sonego 6-4 7-6 (7-4) in almost two hours, while 42nd-ranked Bernabe Zapata Miralles was beaten 6-2 7-6 (8-6) by Ugo Humbert.
Finland's Emil Ruusovuori triumphed with Constant Lestienne retiring down 6-4 4-1 and world number 46 Alexander Bublik was a shock casualty, going down 6-4 6-4 to Wu Ting-Lin.
Murray, ranked 53rd in the world, could not find his usual return with Lajovic winning 72 per cent of second-serve points, prevailing in one hour and 38 minutes.
The Serbian claimed only his fourth win out of 16 matches on hard courts since the start of last year, holding his nerve after failing to serve out the match at 5-4 in the second set, converting his third match point in the 12th game.
Lajovic hit 21-14 winners, with Murray committing more unforced errors (15-13). The Serbian converted all three break points he generated.
Former world number six Gael Monfils was forced to retire due to a right wrist injury in his clash with French compatriot Ugo Humbert at 3-3.
Monfils, 36, was playing at only his third event since returning to the ATP Tour following seven months out due to injury.
Argentina's Facundo Bagnis defeated Brazilian qualifier Felipe Meligeni Alves 6-3 1-6 6-4, with his reward a second-round clash with last week's Indian Wells Open winner and top seed Carlos Alcaraz.
World number 50 J.J. Wolf beat world number 48 Alexander Bublik 7-5 6-3 in 79 minutes, earning a second-round clash with sixth seed Andrey Rublev.
World number 74 Martin Fucsovics sent down nine aces as he beat Argentina's Pedro Cachin 6-4 7-6 (7-2) in 107 minutes. Fucsovics will next face seventh seed Holger Rune.
Ilya Ivashka beat Daniel Altmaier 6-2 6-1 to book a second-round clash with third seed Casper Ruud, while Fabio Fognini bowed out, losing 6-4 5-7 6-4 to Jan-Lennard Struff. USA's Brandon Nakashima powered to a 7-6 (7-3) 6-3 victory over Germany's Oscar Otte.
Nadal – playing his first competitive tournament since the Australian Open – produced some highlight moments as he saw off Miomir Kecmanovic 6-2 7-5 in Acapulco on Wednesday.
A two-time winner of the ATP 500 event, world number two Nadal was a class above against his Serbian opponent to stay on course for the title.
After his powerful display, the 19-time grand slam champion will face Kwon Soon-woo for a spot in the semi-finals after the South African beat eighth seed Dusan Lajovic 7-6 (7-2) 6-0.
Australian Open semi-finalist Zverev was bundled out of the tournament by American qualifier Tommy Paul 6-3 6-4.
Zverev dropped his opening service game and it was a sign of things to come for the German star as Paul capitalised to eventually claim the biggest win of his career.
Next up for 22-year-old Paul is fifth seed John Isner, who downed fellow American Marcos Giron 6-3 7-6 (7-4).
Grigor Dimitrov saved two match points as he prevailed 6-7 (8-10) 6-2 7-6 (7-2) against Adrian Mannarino in a thriller.
Mannarino erased Dimitrov's 4-1 lead in the final set to earn a pair of match points but the Bulgarian rallied to set up a showdown with third seed Stan Wawrinka, who eased past Pedro Martinez 6-4 6-4.
Elsewhere, fourth seed Felix Auger-Aliassime was surprised 6-4 6-4 by Kyle Edmund and Taylor Fritz topped Ugo Humbert 6-4 6-1.
At the Chile Open in Santiago, seeds Casper Ruud, Hugo Dellien and Federico Delbonis all moved through to the quarters but Pablo Cuevas fell to qualifier Renzo Olivo.
Nadal was in devastating form as the 19-time grand slam champion dismantled Kwon 6-2 6-1 at the ATP 500 tournament in Acapulco on Thursday.
After crushing Miomir Kecmanovic in the last 16, Nadal continued his ruthless path through the draw at the expense of the rising South Korean in one hour, 31 minutes.
Nadal – eyeing his third Acapulco title – hit 25 winners, 11 unforced errors and saved all eight break points he faced to set up a semi-final showdown with Grigor Dimitrov.
Bulgarian Dimitrov snapped a five-match losing streak against three-time major champion Stan Wawrinka by defeating the third seed 6-4 6-4.
Dimitrov – the 2014 Mexican Open champion – had not beaten Wawrinka since 2016 but the former world number three ended his drought in one hour, 25 minutes.
"I've played quite a few times against Stan and we've practised together so many times. Between us, it's mainly a mental battle," Dimitrov said. "I had lost the past five times against him, but those losses have inspired me. Those losses helped me. Even though it hurts saying it, I'm admitting it. I wanted to stand tall tonight."
John Isner – the fifth seed – booked his spot in the semis for the second successive year with a 7-6 (7-3) 3-6 6-2 win over fellow American Tommy Paul.
Isner fired down 22 aces to set up a clash against countryman Taylor Fritz, who topped Kyle Edmund 6-4 6-3.
At the Chile Open, in-form top seed Cristian Garin extended his winning streak en route to the quarter-finals in Santiago.
Garin – the Cordoba and Rio Open champion – celebrated his 10th consecutive victory by downing Alejandro Davidovich Fokina 6-2 0-6 7-6 (7-4) at the ATP 250 event.
Fellow seeds Albert Ramos-Vinolas and Thiago Monteiro also progressed but fifth seed Juan Ignacio Londero was upstaged by wildcard Thiago Seyboth Wild 7-6 (9-7) 6-4.
Nadal displayed an abundance of tactical nous, nullifying the American’s big hitting and service game to emerge the 7-6(6-3) 7-6(7-5) winner.
Along with a 76 percent first-serve rate, Opelka hit more winners with 26 for the match, but the 35-year-old Spaniard was able to grind out points from the baseline with his trademark heavy topspin. As a result, Nadal’s winner/unforced error differential was +14 in comparison to Opelka’s +1.
"He is one of the toughest opponents on tour," Nadal said post-match. "It is very tough to control his weapons with his serve and forehand.
"I think I played my best match of the tournament so far today. I am very pleased with how I was able to win the match, with two difficult tie-breaks. This victory means a lot to me."
The highest ranked player left in the draw, Nadal will now face Nick Kyrgios, who progressed to the quarter-finals after Jannik Sinner withdrew with illness.
Matteo Berrettini made a shock exit, meanwhile, losing 6-3 6-7(5-7) 6-4 to unseeded Serbian Miomir Kecmanovic.
The Italian world number six was put under constant pressure, with Kecmanovic targeting his backhand and hovered the baseline to close the angles on serve.
The 22-year-old’s only other top 10 victory came against Alexander Zverev, also the world number six then, at Cincinnati in 2019. He will now face Taylor Fritz, who defeated Alex de Minaur 3-6 6-4 7-6(7-5).
Also on Wednesday, Grigor Dimitrov edged past John Isner 6-3 7-6(8-6). In his unique style, the Bulgarian 33rd seed came up with the shot of the day, flicking a forehand pass across the visibly stunned Isner.
He will face Andrey Rublev, who defeated Hurbert Hurkacz 7-6(7-5) 6-4. In Wednesday’s other results, Carlos Alcaraz Garfia comfortably defeated Gael Monfils 7-5 6-1, while Cameron Norrie accounted for Jenson Brooksby 6-2 6-4.
World number two Nadal, chasing a record-extending sixth title on home soil, advanced 6-3 6-3 in a time of one hour and two minutes on Thursday.
Popyrin, playing in his first last-16 match at a Masters 1000 event, broke Nadal in the sixth game of the opening set but the Spaniard otherwise looked comfortable.
Nadal converted four of his six break points to advance through to the last eight for a 15th time, where Alexander Zverev awaits.
"It was difficult," Nadal said in his on-court interview. "The court today was slippery, it was very fast, very dry conditions. The ball was flying a lot. The beginning was super tough.
"He was hitting every ball and hitting the spots. I'm happy with the victory. It was an important victory, he came here playing well. Every match is tough."
Zverev, who won the competition in 2018, beat British number one Dan Evans 6-3 7-6 (7-3) to set up a meeting with Nadal
However, second and fourth seeds Medvedev and Tsitsipas saw their participation come to an end at the hands of Cristian Garin and Casper Ruud respectively.
Medvedev, who is still yet to reach the quarters in Madrid, lost the only break of serve in the third game of the opening set but recovered by taking the second set in a tie-break.
Garin dominated the deciding set, the world number 25 breaking Medvedev twice and, after a short delay to fix a broken net, advanced 6-4 6-7 (7-2) 6-1.
Tsitsipas, who held championship point against Nadal in the Barcelona Open final, failed to break Ruud on a single occasion as he fell 7-6 (7-4) 6-4.
Alexander Bublik is next up for Ruud after beating Aslan Karatsev 6-4 6-3, while Matteo Berrettini - 7-6 (7-4) 6-4 winner against Federico Delbonis - will meet Garin.
Also vying for a place in the semi-finals are John Isner and Dominic Thiem, who are set to face off after beating sixth seed Andrey Rublev and Alex de Minaur respectively.
The world number four's bid for an 11th title in Rome got off to a strong start on Wednesday as he saw off big-serving Isner 6-3 6-1 in a time of 76 minutes.
Isner twice missed the chance to break Nadal in the seventh game of the opening set and the Spaniard took control from that point on in the second-round match.
Nadal, who has won this tournament in three of the past four years, identified that hold of serve as a key point in the contest.
"The beginning of the match was not good for me," he said in his on-court interview. "He had some chances on the return and had two break points.
"He had two not difficult balls so I was in his hands at that moment. I was lucky that he missed those shots and then I was able to break.
"Then the match changed, of course. With the first set on the board, and having the break in the first game of the second set, everything changed."
Nadal will now face Shapovalov in a repeat of last year's last-16 encounter, which the record 21-time grand slam winner edged in three sets.
He recovered from a set down and saved two match points before beating the Canadian 3-6 6-4 7-6 (7-3) en route to lifting the title in the Italian capital.
And Nadal, who was beaten by Carlos Alcaraz on clay in last week's Madrid Open quarter-final, is not expecting an easy task this time around.
"Last year was a joke, the match that I saved here against him," said Nadal of his next opponent. "I was super lucky. I know how dangerous he is, I need to play well.
"I need to play better than today, but after a while without being on court it is another victory and I have the chance again to play against one of the best players in the world.
"I need to build things again after a tough stoppage and that's what I am trying now. I just need to stay with the right attitude, and let's see if I am able to make that happen."
Alexander Zverev also booked his place in the last 16 on Wednesday thanks to a 7-6 (8-6) 6-3 victory over Sebastian Baez.
Last week's Madrid Open runner-up was given a tough time of things by in-form Baez, but ultimately came through unscathed to stay on course for more silverware.
Top seed Nadal was too good for Dimitrov en route to his fourth Mexican Open decider, triumphing 6-3 6-2 in convincing fashion at the ATP 500 tournament on Friday.
Nadal won the event in 2005 and 2013 and the 19-time grand slam champion has the chance to claim a hat-trick of Acapulco titles after maintaining his relentless form.
"I'm very happy. A great victory against one of the best players in the world and a good friend," Nadal said. "I think I've increased my level during the match, so it's positive for me."
Spanish star Nadal recovered from 1-2 start, reeling off four consecutive games with some powerful baseline winners to seize control against seventh seed Dimitrov.
It was a similar theme in the second set as Dimitrov broke for a 2-0 lead but that was as good as it got for the Bulgarian – Nadal going on a six-game run to close out proceedings in one hour, 41 minutes.
Awaiting world number two Nadal in the final is Taylor Fritz after the American stunned fifth seed John Isner 2-6 7-5 6-3 in a stunning comeback.
"He's playing well. He has a great serve, fantastic shots from the baseline. It's going to be a tough one," Nadal said. "I know I'm going to have to be at my best and I hope to be ready for it."
At the Chile Open, second seed Casper Ruud defeated seventh seed Federico Delbonis 7-5 7-5 to book his spot in the semi-final.
Ruud – the first Norwegian to win an ATP Tour title in Buenos Aires – will next meet third seed Albert Ramos-Vinolas, who saw off Thiago Monteiro 6-4 6-4 in Santiago.
Top seed Cristian Garin was forced to retire due to a leg injury after losing the first set 7-6 (7-1) to Thiago Seyboth Wild, while Renzo Olivo upstaged Pablo Cuevas 3-6 7-6 (7-5) 7-5.
Nadal, a five-time champion at the event, was a break up in the first set of his 15th quarter-final at Caja Magica but lost 6-4 6-4 in an hour and 44 minutes.
The win was the biggest of Zverev's career on clay and confirmed his rise to second in the ATP Rankings.
While Nadal said he was largely pleased with his displays in the Spanish capital, he was at a loss to explain the manner of his defeat on Friday.
"I'm leaving Madrid with an overall positive feeling, but at the same time with the ugly feeling of having played a match like this today against a great player," Nadal told a media conference.
"I think for most of the first set, I was playing better than him. And this is the negative part; while playing better than him in the first set, I still lost 6-4.
"This is very difficult to understand, especially for me. When this happens to me, it's usually in the opposite way. I find a way to win sets even though I'm not playing my best or as good as the opponent.
"I felt like I was playing better for much of the first set, but after a couple of errors – unjustifiable errors at the worst times – I found myself down a set.
"The outlook of the match changed there, both for me and for him. I knew then that a lot of suffering was waiting for me [in the second set], and for him, the knowledge that he just took a huge step forward during the match."
Zverev will face Dominic Thiem in the last four after the Austrian overcame John Isner 3-6 6-3 6-4 in an hour and 55 minutes.
Thiem, who has now made it to the past four Madrid Open semi-finals, said: "I think we all know that he is one of the best servers in history and the altitude here in Madrid makes it even tougher to return his serve.
"I was a little bit surprised by his return games. I think he actually attacked both of my serves, the first and the second, and it took me a while to get used to it."
Casper Ruud will meet Matteo Berrettini in the other semi-final.
Norwegian Ruud eased past Alexander Bublik 7-5 6-1 to make it three consecutive clay ATP Masters 1000 semis, while Berrettini beat Cristian Garin 5-7 6-3 6-0.
The third-seeded Norrie defeated Kyrgios 6-1 6-4 in less than an hour as he tries for a second consecutive title after collecting his first ATP Tour championship at Los Cabos last week.
The former University of Georgia star Isner, a five-time champion in Atlanta, beat Sock 7-6 (8-6) 6-4 despite serving only 13 aces after hitting 36 in his previous match.
Second-seeded Jannik Sinner fell 7-6 (9-7) 6-4 to Australian qualifier Christopher O'Connell, leaving the tournament without its top two seeds after Milos Raonic was upset by Brandon Nakashima on Wednesday.
O'Connell, ranked 132 in the world to Sinner's 23, had not won a main-draw match this year before defeating Denis Kudla in the opening round and will now face Isner in the quarters.
Norrie will meet Emil Ruusuvuori, who advanced when Benoit Paire retired down 3-0 in the third set after the pair had split the first two sets 4-6 6-4.
Ruusuvuori is into his second career ATP quarter-final after making it to the semis at Nur-Sultan last year.
Opelka kept up his strong early-season form with an imposing 6-3 7-5 victory over Nick Kyrgios in the first semi-final, getting the job done in just one hour and nine minutes.
Opelka slung down 18 aces to 11 from Kyrgios and faced only one break point across the contest.
The 24-year-old Opelka won a title in Dallas in February and followed up with a run to the Delray Beach final, with a third trophy match of the season now awaiting him.
The American world number 18 said he admired the "intangibles" from Kyrgios that "you can't teach."
"He's got one of the best serves in the world and when he turns it on he can come up with some shots that not many guys outside the top 10 can," Opelka said in his on-court interview.
Opelka enjoyed the high-tempo match with Kyrgios, who typically wastes no time between points.
"I prefer to play at that speed rather than, for example, Rafa [Nadal] where it's 50-55 seconds between points," Opelka said.
"It's a totally different speed here and I thought I did a good job managing the whole tempo of the match. I put a lot of returns into play and I came up big with my second serve."
He will face 2013 Houston champion Isner, who came from a set down to defeat defending champion Cristian Garin 4-6 6-3 6-4.
The 36-year-old's serve improved as the match went on and ultimately propelled him to victory, facing four break points in the opening set, one in the second and none in the decider.
Isner also served 17 aces on his way to the final, and won 80 per cent of points on his first serve.
"It's never easy when you have to come back from one set down," he said. "My serve certainly helped me out a lot. I didn't have to hit too many balls there in the end, and I didn't want to.
"I'm so happy to be in the final again here. At my age you never really know when you can get back to a final of a tournament like this."
The giant American, who measures in at six-foot-11, was terrific in the opening set, winning 84 per cent of his successful first serves (16-19) and allowing no break point opportunities.
The second set was much more competitive, as Brouwer broke first to go up 3-1, but Opelka answered instantly and then grabbed the match-winning break when 6-5 up to avoid a tie-breaker.
Opelka will face a fresh Nick Kyrgios in the semi-final after the Australian was gifted a walkover win against Michael Mmoh due to injury.
In a clash between two top-five seeds, world number 29 Christian Garin got the better of world number 13 Taylor Fritz 6-2 6-7 (5-7) 6-3.
What made Garin successful in the first and third sets was his return game, winning more than half of his return points in those sets combined (26-51).
Fritz had chances to get back into the game, but Garin saved 11 of 14 break point opportunities, including six of seven in the deciding third set.
Garin will play American John Isner in the semi-final after Isner triumphed in a three-set battle against Frances Tiafoe 6-4 2-6 6-3.
Isner's trademark serve was inconsistent, and abandoned him in a grim second-set performance, winning just 11 of 21 service points on his way to getting broken twice to allow Tiafoe to force the match into a third frame.
In Isner's two successful sets, he won more than 80 per cent of his first serves (35-41), and allowed no break point opportunities in the third.
Ruud, a French Open finalist and defending champion in Bastad , lacked his usual composure as world number 39 Cerundolo broke him three times to win 6-4 3-6 7-5 on Wednesday.
That marked a first top-10 career victory for Cerundolo after defeating world number five Ruud, guiding him to his first quarter-final appearance at the tournament.
"I'm feeling amazing, I'm really happy," said Cerundolo, whose previous best result was beating world number 24 Gael Monfils. "It's amazing getting my first Top 10 win. It's been the best year so far in my career.
"He's a French Open finalist and was the defending champion here, so it was a big challenge for me and I'm happy to go through."
Aslan Karatsev awaits in the quarter-finals Cerundolo after downing Frenchman Hugo Gaston 6-2 6-4.
Fifth seed Pablo Carreno Busta survived a first-set scare to defeat 2016 champion Albert Ramos-Vinolas 4-6 7-6 (7-5) 7-5.
The victory was Carreno Busta's 250th tour-level triumph and he will next face third favourite Diego Schwartzman, who overcame Elias Ymer in straight sets.
Third seed Alexander Bublik was made to work for a 6-3 3-6 6-2 second-round victory over Jack Sock in Newport and Andy Murray edged past Wimbledon doubles champion Max Purcell.
Murray needed almost two-and-a-half hours to get the better of Purcell, eventually triumphing 4-6 6-2 6-1 to tee up a last-eight meeting with Bublik.
Fourth favourite Maxime Cressy had little trouble in beating fellow American Mitchell Krueger 6-3 6-4 and will next face Steve Johnson, who progressed past seventh seed Jiri Vesely with a walkover.
Benjamin Bonzi, ranked as fifth seed in the United States, eased past Christopher Eubanks in straight sets, while James Duckworth triumphed 7-5 3-6 6-2 over Quentin Halys.
John Isner joined the list of favourites who made the third round in routine fashion, beating Peter Gojowczyk in straight sets 6-3 7-6 (8-6), before top seed Felix Auger-Aliassime faced some adversity against Jason Kubler following the Australian's strong run into the fourth round of Wimbledon.
Auger-Aliassime has a slight advantage, leading 6-4 3-6 6-5 before play was suspended due to poor light. When play resumes, Kubler will serve with a chance to force a third-set tiebreak.
