Team World went into the final day of action at the O2 Arena in London sitting four points behind their opponents, but they produced a stirring fightback to claim the trophy at the fifth time of asking.
Central to their success was Canadian Auger-Aliassime, who beat Novak Djokovic in singles after successfully teaming up with Jack Sock in the doubles.
Holding an 8-4 lead from Saturday, many expected Team Europe to breeze it from there, but John McEnroe's World team had other ideas and earned a 13-8 victory.
Up first in the doubles were Auger-Aliassime and Sock, who lost the first set to Team Europe's Andy Murray and Matteo Berrettini.
However, the World duo roared back to deliver three points for their team by claiming a 2-6 6-3 10-8 victory.
Djokovic won two matches for Team Europe on Saturday, yet he was powerless to stop Auger-Aliassime in their singles clash. The 22-year-old Canadian landed a 6-3 7-6 (7-3) over the Wimbledon champion.
That moved Team World 10-8 ahead in the overall contest, setting the stage for a decisive clash between Stefanos Tsitsipas and Frances Tiafoe, with a further three points at stake.
Greek Tsitsipas won the first set, but 24-year-old American Tiafoe stormed back to win an epic tie-break in the second on his way to a 1-6 7-6 (13-11) 10-8 success.
Initially a wildcard for the finals as the highest-ranked losing nation in the qualifiers, the 2019 runners-up are through to the last four for the second time in three years - and fourth time overall.
Felix Auger-Aliassime overcame Oscar Otte in straight sets - condemning him to a fourth straight singles defeat - to set up a doubles decider and cancel out Denis Shapovalov's earlier loss to Jan-Lennard Struff.
Shapovalov and Vasek Pospisil appeared in trouble when they lost the opening set 6-2 against Tim Puetz and Kevin Krawietz, who had won all eight Davis Cup matches when paired together.
But the Canadian duo, who hit a combined 11 aces during the contest, rallied to force a deciding set, which they took 6-3 to inflict a first defeat on their opponents.
Earlier in the day, Italy also came through a deciding doubles match to beat the USA and secure their first semi-final appearance since 2014.
Despite missing the injured Jannik Sinner and Matteo Berrettini, the Italians were appearing in their first quarter-final for six years, and they made a strong start with Lorenzo Sonego hitting 17 aces on the way to defeating Frances Tiafoe 6-3 7-6 (8-6).
Targeting only their fourth semi-final since 2007, 32-time winners USA rallied with Taylor Fritz overcoming Lorenzo Musetti in straight sets to force the first deciding double match of the finals.
There, just one break per set was enough for Simone Bolelli and Fabio Fognini to beat Tommy Paul and Jack Sock 6-3 6-3, and send the 1976 champions through.
Djokovic had not played since taking his grand slam count to 21 with the Wimbledon title in July, but he looked as if he had hardly been away as the Serbian beat Frances Tiafoe and then teamed up with Matteo Berrettini in the doubles.
Tiafoe beat Rafael Nadal at the US Open earlier this month and combined with Jack Sock on Friday to defeat the Spaniard again in what was Roger Federer's last ever match, but the American could not get to grips with Djokovic in Saturday's final singles contest.
Djokovic's 6-1 6-3 win secured two points for Team Europe to put them out in front, and he was involved again in the last of the day's action as the team's advantage doubled to four points.
Berrettini and Djokovic tussled with Sock and Alex de Minaur and ultimately had too much, winning 7-5 6-2 in less than an hour and a half.
Earlier, Taylor Fritz's three-set triumph over Cameron Norrie put Team World briefly back on level terms.
Fritz made a brutal start but ended up being forced to a match tie-break, eventually coming through with a 6-1 4-6 10-8 victory.
That wiped out the two-point lead Berrettini had given Europe in the first match of the day, with the Italian edging Saturday's most gruelling tussle.
He saw off Felix Auger-Aliassime 7-6 (13-11) 4-6 10-7, and despite finding himself on court for over two hours, Berrettini was still sharp enough to emerge victorious alongside Djokovic.
Belarus' Ivashka is ranked over 50 spots higher than Sock, but he is in some torrid form, with Monday's loss his sixth in a row.
Sock snapped his own three-match losing streak, and he did so despite committing 10 double faults, coming up clutch in break point situations by saving 15 of Ivashka's 18 opportunities.
Meanwhile, 21-year-old Argentinian Juan Manuel Cerundolo booked a meeting with compatriot and top seed Diego Schwartzman after beating Pablo Andujar 6-4 1-0 (retired) in the first round of the Cordoba Open.
The rising Cerundolo now boasts a 13-1 record since the beginning of the new year, while it was Andujar's fifth loss in a row.
In an all-Spanish showdown it was Bernabe Zapata Miralles emerging victorious 6-7 (6-8) 6-3 7-5 against Roberto Carballes Baena, while Serbia's Dusan Lajovic defeated Peru's Juan Pablo Varillas 7-6 (7-3) 6-3.
The last match of the night saw Italy's Luciano Darderi handle France's Hugo Gaston 6-4 6-3, with Darderi now set to face fourth seed Sebastian Baez.
Kyrgios' replacement in Adrian Mannarino eventually defeated Gojowczyk 6-3 7-6 (9-7), but the Wimbledon finalist's withdrawal due to injury overshadowed the match.
After receiving a wild card for the Cincinnati Masters, the 2016 champion in Atlanta was expected to make a deep run to kick off his hard-court season.
Coming out to apologise to the crowd before Gojowczyk and Mannarino took to the court, Kyrgios expressed his sadness at the withdrawal, in the hope he could still play in the doubles draw with fellow Australian Thanasi Kokkinakis.
"I'm extremely shattered that I'm not able to compete tonight," he said pre-match. "I've won this tournament once, and you know, I'm playing some of the best tennis of my career.
"All I wanted to do was come out here and give you guys a show, to see what I was capable of. I'm unable to give out my best performance today, and I'm just extremely sorry.
"I'm going to keep my hopes up and maybe be able to continue doubles with Thanasi this week. I hope you'll all not be too hard on me."
Kokkinakis just has the doubles to attend to now after Andres Martin's 6-3 6-2 win, while Ilya Ivashka progressed with a 6-0 3-6 6-3 win over Quentin Halys.
James Duckworth was the lone Australian to advance on Tuesday, beating Dominik Koepfer 6-1 6-7 (5-7) 6-3.
Jenson Brooksby made an encouraging Atlanta Open debut earlier in the day, eliminating Benoit Paire 6-3 6-1 to reach the second round.
Countryman and fifth seed Tommy Paul breezed past wild card Jack Sock 6-1 6-1, while Ben Shelton and Mackenzie McDonald also advanced.
Murray, who had not played at Indian Wells in four years, won in one hour and 24 minutes with a disciplined display, winning 82 per cent of first serve points.
The Briton, who only faced one break point for the match, will next take on Spanish 18-year-old Carlos Alcaraz, who enjoyed a fine run to the US Open quarter-finals in September.
"I think Alcaraz is really, really good," Murray said after the win. "I think he’s got a good chance at it."
On his own performance, Murray added: "I thought my mentality was good tonight. I sort of focused just on every single point.
"It's been a tough road to get back and it was so nice that so many people managed to come out and support today. I really enjoyed it."
BROOKSBY SETS UP ZVEREV DATE
Emerging American Jenson Brooksby triumphed over Cem Ilkel 7-6 (7-5) 6-4 to book his spot in the second round where he will face third seed Alexander Zverev.
The 20-year-old world number 79, who was the runner-up at the Hall of Fame Open in July, prevailed in his Indian Wells debut in one hour and 44 minutes.
Brooksby dropped only four points on his serve, with Ilkel unable to generate a break point throughout the match.
The California native has been a break-out star in the latter half of 2021, reaching the fourth round at the US Open, entering the top 100 after starting the year ranked 307th.
SOCK PUT AWAY BY AUSTRALIAN
Former top 10 player Jack Sock was bundled out by Australian John Millman 7-5 4-6 6-3 in an epic first-round clash which lasted two and a half hours.
Sock, who has battled injuries in recent times and was handed a wild card for Indian Wells, pushed Millman all the way, with the Australian breaking to lead 5-3 in the third set before serving out for victory.
World number 61 Pedro Martinez made light work of Italian qualifier Roberto Marcora 6-4 6-0 to set up a second-round meeting with second seed Stefanos Tsitsipas. The 24-year-old Spaniard triumphed in one hour and 15 minutes.
Former Wimbledon and US Open finalist Kevin Anderson sent down 10 aces in a 7-5 6-2 win over Jordan Thompson.
In the singles, Kyrgios' serve was overpowering against his Japanese opponent, winning 76 per cent (38-of-50) of his points on serve, and saving the only break point opportunity he faced, which came early in the opening set.
His return game was also sharp, collecting breaks of serve in the first game of each set to snatch all the momentum and keep Nishioka on the back foot and chasing all day.
Kyrgios finished with 12 aces to Nishioka's three, and finished the match just like he started it with a break of serve.
It caps off a tournament where he dropped only one set in his six matches – against Frances Tiafoe in the quarter-final – to win his first singles title since this same event back in 2019.
Just a few hours later, Kyrgios was back on the court with Sock contesting the doubles final, with the pair saving both break points they faced.
After an even start, Kyrgios and Sock broke late in the first set, and then struck again to break in the opening game of the second set, going on to serve things out comfortably, never allowing Dodig and Krajicek to even reach deuce against their serve.
In doing so, Kyrgios became the first player to ever sweep both the singles and doubles in Washington.
He only had one doubles title before this season – back at the Lyon Open in 2018, also partnered with Sock – but Kyrgios has now won three in the past eight months, emerging triumphant with fellow Australian Thanasi Kokkinakis at both the Australian Open and last month's Atlanta Open.
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.
The final event of Federer's playing career veered off the script as he and Rafael Nadal lost in doubles on Friday, before the team collectively succumbed to a 13-8 defeat in London.
Stunning singles wins for Felix Auger-Aliassime and Frances Tiafoe on Sunday, against Novak Djokovic and Stefanos Tsitsipas respectively, followed a doubles thriller that saw Auger-Aliassime and Jack Sock topple Andy Murray and Matteo Berrettini.
After Team Europe won the first four editions of the Laver Cup, this time they had to swallow the bitter pill of defeat, with Federer sorry to sign off on a losing note.
"Of course I'm disappointed," he said. "I was on the team. I almost lost my voice. My hands hurt from clapping.
"So, yes, I am disappointed. We wish the result would be different. I told Andy [Murray] in the locker room, I don't like losing. It's not fun. It just leaves not the best taste, you know. I think once you have been there and taste success, it's just not the same."
He said his goodbye tournament had been a mix of highs and lows.
"This weekend has been all over the place for me," said the 41-year-old Swiss. "I enjoyed it, but it's unfortunate that we couldn't get the win tonight."
Federer denied he has his eyes set on becoming the next Team Europe captain. Incumbent skipper Bjorn Borg and Team World counterpart John McEnroe have indicated next year's match could be the last that they helm, which would create an appealing vacancy.
"No plans there. Bjorn's doing a great job," said Federer. "Who knows, maybe one day, but we don't have any plans so far."
Next year's match takes place in Vancouver, and Federer will certainly have a role of some sort to play, given he is a co-founder of the event.
"I went through all different types of Laver Cups so far: the first one, the winning teams, now this time on the losing team," Federer said. "There was also one where I was hurt last year but seeing it more from the stands and from the fans' perspective, and now deep on the inside with retirement.
"I have enjoyed the Laver Cup in many different ways, and next year again will be totally different. I'm looking forward to it, and I'm sure Vancouver is going to be fantastic."
Asked what he would miss about tennis, Federer said: "Not the losing press conferences, I tell you that. They are the worst."
The Croatian came through qualifying to make his ATP Tour main-draw debut in Florida and the 33-year-old is through to the last 16 after a 4-6 6-2 6-2 victory against former world number eight Sock on Tuesday.
World number 784 Pecotic has a full-time job in finance and he will be in the money after setting up a meeting with Marcos Giron in the last 16 of the ATP 250 event.
The Princeton and Harvard graduate's boss was in his box to watch him see off Sock and Pecotic hopes he will get a chance to book a quarter-final spot on Wednesday rather than return to balancing the books.
He quipped: "I had to leave work early today. I had to send an email to the whole team. [My boss] let me off. I'm going to have to ask for another day off tomorrow."
Pecotic, who served 10 aces and struck 30 winners, added: "I certainly didn't expect to win, but certainly didn't come into the match thinking that I'm definitely going to lose.
"You've got to be realistic. This is a former top 10 guy with an incredible amount of tennis experience, with a huge serve.
"He came out serving 134 [mph] on the first serve. It would be arrogant to think that I'm going to come out and expect to win.
"But I certainly figured if I could sink my teeth into the match and work on the two or three patterns that I prepared before, that I'm going to have a chance. And then let's see... if I was going to fall under pressure or not be able to serve it out at some point. But I didn't and I got the win."
Now unranked, Sock needed a wildcard into the ATP 250 tournament and suffered a 6-3 6-4 loss to fellow American Marcos Giron.
A former world number eight, Sock won just one singles match last year – at the Laver Cup.
Giron created 10 break points and converted four on his way to victory in 76 minutes.
Coming off a run to the Australian Open quarter-finals, Sandgren suffered a 6-7 (5-7) 6-3 7-6 (7-3) loss to Steve Johnson.
Sandgren squandered seven match points in a last-eight loss to Roger Federer in Melbourne and the fifth seed was edged by Johnson.
Sixth seed Miomir Kecmanovic and Italian veteran Andreas Seppi posted wins over Tommy Paul and Damir Dzumhur respectively.
At the Argentina Open, Casper Ruud was the only seed in action and eased past Pablo Andujar 6-2 6-3.
Pablo Cuevas, Federico Delbonis and Thiago Monteiro also moved into the second round.
Federer teamed with long-time rival Rafael Nadal against American duo Frances Tiafoe and Jack Sock as Team Europe face Team World at the O2 Arena.
After winning the first set, the Swiss and the Spaniard were faced with a spirited fightback from Tiafoe and Sock, who won a second-set tie-break before also clinching the match tie-break to seal victory 4-6 7-6 (7-2) 11-9.
There had been a great atmosphere in London on Friday for the fifth edition of the three-day competition.
Federer, 41, had not played a match since he was knocked out of Wimbledon by Hubert Hurkacz at the quarter-final stage last year due to knee injury.
There were huge cheers when Federer and Nadal came onto the court and loud roars for the Swiss great when he came up with his first winner.
There was a sensational moment in the first set when Federer appeared to have won a point after his shot hit the top of the net and bounced in, only for a replay to show the ball had somehow been hit through a small hole between the net and the post, meaning Team Europe lost the point, despite the remarkably unlikely event of the ball passing through such a gap.
Novak Djokovic and Andy Murray were among Federer and Nadal's team-mates offering some tactical input between games, and the legendary duo had to save a first break point of the match before the Spaniard held to put them 5-4 up.
It was two of the all-time greats that took a tight opening set when Sock put a volley into the net, prompting Team World captain John McEnroe to tell his doubles pair they appeared to be getting "caught up in the hoopla."
A determined Team World managed to level it up after a back-and-forth second set went to a tie-break, with Tiafoe and Sock coming out on top with their superior power and pace.
That took it to a dramatic deciding match tie-break, with every point keeping fans holding their breath, including some trademark Federer magic at 6-7 when he diverted a Sock shot across court for a crowd-pleasing winner.
It looked like the dream script was being followed as Federer served at match point for himself and Nadal, but the former could only hit a shot into the net, before Sock and Tiafoe won the next two points to secure the win for Team World.
It was then hugs all round as respect was paid to the 20-time grand slam winner at the conclusion of a phenomenal career.
Earlier in the day, Casper Ruud drew first blood for the Team Europe, beating Sock 6-4 5-7 10-7, before Stefanos Tsitsipas doubled their advantage with an emphatic 6-2 6-1 defeat of Diego Schwartzman.
There was drama after the first set of that second singles match when a protester, wearing a T-shirt with the message, "END UK PRIVATE JETS" on, set their arm alight on court before being escorted out by security.
Alex de Minaur then got Team World on the board with a 5-7 6-3 10-7 success over home favourite Murray before the late-night main event under the lights.
Thiem, last in action in an Australian Open final loss to Novak Djokovic, returned with a 6-2 4-6 6-1 victory over wildcard Felipe Meligeni Alves on Tuesday.
The 2017 champion of the ATP 500 event played on clay in Brazil, top seed Thiem grabbed two breaks of serve in the deciding set.
Having started 2020 with four losses in five matches, Borna Coric found a victory by overcoming Juan Ignacio Londero 7-6 (7-5) 7-5.
Dusan Lajovic joined Thiem and Coric as the three seeds to win, but two fell in the opening round.
Casper Ruud, coming off a title win at the Argentina Open, went down to qualifier Gianluca Mager 7-6 (7-4) 7-5.
Guido Pella, the 2016 runner-up, was edged by Thiago Monteiro 5-7 6-4 7-6 (7-3) after just over three hours.
At the Delray Beach Open, Sock claimed his first non-Laver Cup singles win since late 2018.
The American 2017 champion, a wildcard, battled past last year's winner Radu Albot 3-6 6-3 7-6 (7-2), saving a match point in the final set and breaking down in tears following his victory.
Canadian second seed Milos Raonic brushed past lucky loser Denis Istomin 6-2 6-2 and Reilly Opelka served 20 aces in a 6-7 (5-7) 6-4 7-6 (7-4) victory over Ernests Gulbis.
Ugo Humbert, the French sixth seed, battled through, while third seed Taylor Fritz fell to qualifier Cameron Norrie 6-4 6-7 (4-7) 6-4.
The fifth seed was taken all the way to a third-set tie-break by Sock, which he trailed 5-3 before recovering to claim a 7-6 (7-5) 3-6 7-6 (7-5) victory in a gripping contest.
Sock, ranked 140 places below his opponent, put his strong forehand to good use to hold in a first set that went the distance but was undone by some unforced errors.
After responding brilliantly by taking the second set, after once again holding his serve, the American looked to be heading for defeat when 40-0 down at 5-6 in the third set.
However, he dug deep to force another tie-break and was within two points of a big victory, only for two more unforced errors to cost him a place in the last 32.
"It was a crazy battle; we both left everything out there," Tsitsipas said. "Jack played incredibly well at times but I was able to bring out the best in my game at the end.
"I proved I could play aggressive tennis and stay calm at the same time."
Tsitsipas will now face another home hopeful in Jenson Brooksby, who saw off Karen Khachanov 6-0 6-3 earlier in the day.
Jannik Sinner battled to a 6-3 6-3 win over Laslo Djere elsewhere in Saturday's action, while Denis Shapovalov and defending champion Cameron Norrie also advanced.
Fabio Fognini withdrew from his meeting with Nikoloz Basilashvili, who now awaits Norrie in the next round.
Brooksby, who received a wildcard for the draw, triumphed in one hour and 49 mnutes against the former world number five.
The 20-year-old American saved a set point at 5-6 in the first set, before winning in a tiebreaker off the back of winning six consecutive points having trailed 1-4.
Former City Open finalist Anderson, who sent down 10-1 aces, had defeated Brooksby to win the Hall of Fame Open last month.
World number 130 Brooksby, who had started the season outside the top 300, claimed the crucial break in the eighth game of the second set.
Brooksby advances to take on American Francis Tiafoe, fresh from competing at Tokyo 2020, as numerous big names enter the draw for the second round.
Kei Nishikori, who won in Washington in 2015, got past American Sam Querrey 6-4 6-3 to set up a second round meeting with Kazakhstani ninth seed Alexander Bublik.
American Jack Sock will take on top seed Rafael Nadal in the second round after he got past Yoshihito Nishioka in a walkover 6-7 (3-7) 4-0.
Former top 20 player Andreas Seppi triumphed in three sets against Yasutaka Uchiyama while Canadian tour veteran Vasek Pospisil got past Emilio Gomez in three.
Ilya Ivashka, Marcos Giron, Ricardas Berankas and Daniel Galan were the other first round winners on Monday.
