The highly rated Spanish teenager eventually downed the German wildcard 3-6 6-1 7-6 (7-3).
Kuhn, ranked 259 in the world, was aggressive early on against Alcaraz, and was rewarded with the first set.
But the 19-year-old – playing as top seed in an ATP Tour event for the first time – rallied to ease through the second set, before being made to work much harder to clinch the win on a tie-break.
Alcaraz will now play Filip Krajinovic in the second round after the Serbian also won in a third-set tie-break against Sebastian Baez 6-1 4-6 7-6 (8-6).
Three seeded players crashed out on Tuesday in straight sets, with third favourite Diego Schwartzman losing against Emil Ruusuvuori 7-5 6-4, sixth seed Nikoloz Basilashvili beaten by Aslan Karatsev 6-4 6-0, and eighth seed Holger Rune going down 7-6 (10-8) 7-5 to Tallon Griekspoor.
Fourth seed Pablo Carreno Busta eased through against Luca Nardi 6-2 6-1, while there were also wins for Fabio Fognini, who sealed his 400th career victory, as well as Daniel Elahi Galan, Borna Coric and Francisco Cerundolo.
At the Swiss Open in Gstaad, sixth seed Cristian Garin lost 6-3 6-4 to Yannick Hanfmann and seventh favourite Hugo Gaston fell to Dominic Thiem despite winning the first set, losing 1-6 6-1 7-6 (9-7).
Elsewhere, Frenchman Richard Gasquet beat Roberto Carballes Baena 7-5 6-4, while his compatriot Benoit Paire retired hurt when a set and a break down against Elias Ymer.
A tight game between Swiss pair Dominic Stricker and Marc-Andrea Huesler saw the former prevail 6-7 (2-7) 7-6 (7-2) 6-4, and an all-Spanish affair was similarly close as Jaume Munar defeated Bernabe Zapata Miralles 6-3 3-6 7-5.
There were also wins for qualifiers Juan Pablo Varillas and Nicolas Jarry against Lorenzo Sonego and Thiago Monteiro respectively.
Murray sped to a 6-3 6-1 victory over Norwegian Viktor Durasovic, a player ranked a lowly 345th by the ATP who won through two qualifying rounds to reach the main draw.
The Scot broke four times across the contest and won 25 of 26 points when landing a first serve, an encouraging rate. Realistically, anything other than a comfortable win for the former world number one would have been concerning ahead of the Australian Open.
Murray at least put last week's disappointing first-round loss to Facundo Bagnis at the Melbourne Summer Set tournament behind him, and now a stiff tests await him in the last-16 stage.
Georgian second seed Nikoloz Basilashvili is next for Murray on Wednesday, with the world number 23 a familiar foe after these two met in the first round at Wimbledon last July.
On that occasion, Murray won in four sets following a desperate wobble in the third, which he lost from 5-0 ahead.
Five times an Australian Open finalist, Murray believes limbering up this week in Sydney can set him up well for the season's first grand slam.
"It is perfect conditions here," Murray said. "It feels like you are playing indoors. It is really nice to get your range here, and hopefully I can carry that through to Melbourne.
"I was hoping to get matches in Melbourne last week, but that didn't happen. Thankfully the tournament organisers and Tennis Australia agreed to give me the wildcard and the opportunity to play here, and I am very grateful for that."
Tuesday also saw a win for Maxime Cressy, the American serve-volleyer who caused a stir with his run to last week's final in Melbourne before pushing eventual champion Rafael Nadal in the title match.
Cressy, who was born in Paris, is a throwback in an era dominated by baseliners, and his game again came good as he saw off Adrian Mannarino 6-3 7-5. Dusan Lajovic awaits him in round two after the Serbian beat Spain's Alejandro Davidovich Fokina 7-6 (7-5) 6-4.
At the Adelaide International 2 tournament, South African fifth seed Lloyd Harris made a first-round exit, losing 5-7 7-5 6-4 to Soonwoo Kwon.
There was another positive performance from Adelaide's own Thanasi Kokkinakis, who swatted away Frenchman Benoit Paire 6-4 6-0 to tee up a second-round shot at American second seed John Isner.
Kokkinakis, a once-bright young prospect who is finding his way again on tour after injury trouble, reached the semi-finals of last week's Adelaide tournament and is eyeing another run in South Australia before heading to Melbourne.
"After a good run last week, I was pretty knackered – I'm not going to lie," Kokkinakis said, quoted in The Australian. "But I thought another one in my hometown was a perfect situation.
"It was kind of a win-win for me almost. If I won, perfect, I get more matches. If not, I get to Melbourne and get to prepare for Aussie Open, but I'm stoked with the win."
The three-time grand slam winner edged 21st seed Basilashvili 6-1 3-6 6-4 6-7 (5-7) 6-4 in a first-round match that lasted three hours and 52 minutes.
It is Murray's first win at the tournament in five years in what was his first outing at Melbourne Park since 2019, when he thought he might have to retire.
Murray was playing on the same court where a retirement video was played after defeat to Roberto Bautista Agut three years ago.
But the 34-year-old has battled back from injury setbacks admirably and last week reached his first ATP Tour final since October 2019 at the Sydney Classic.
With Japan's Taro Daniel now standing between Murray and a place in round three of the Australian Open, the five-time beaten finalist is eager to make up for lost time.
"It's amazing to be back," Murray said in his on-court interview. "It's been a tough three, four years.
"I have put a lot of work to be back here and I have played on this court many times and the atmosphere has been incredible.
"I have always had fantastic support and this is the court I thought I potentially played my last match on.
"But it is good to be back, winning a five-set battle like that. I could not ask for any more.
"I would love to have a deep run here if possible. It's something I have not had at one of the slams since I came back from the injury and it is something that motivates me."
Wild card Murray broke hard-hitting Basilashvili nine times on John Cain Arena, but he looked physically drained as the match dragged on.
The former world number one showed incredible resolve to take the deciding set, however, against an opponent that had lost just once in seven previous five-set battles.
"I will hopefully keep improving. There are things in my game I can definitely do better," Murray said.
"I have played some of my best tennis here over the years. I feel comfortable here and I hope I can do well here this tournament."
The Georgian completed an impressive week in Munich by beating Jan-Lennard Struff 6-4 7-6 (7-5) to clinch the trophy without dropping a set in his five matches.
The win continued what has been a solid 2021 for Basilashvili, who won the title in Doha in March after defeating Roger Federer in his second match following his return from injury.
After retiring from his opening match in Monte Carlo last month before losing to Jeremy Chardy in Barcelona, the world number 35 had looked in fine form in Munich before battling to victory against Struff in just under 86 minutes.
"It wasn't easy and I was super-tight at the end," said Basilashvili. "It was very difficult. It's never easy playing Jan, he's a very big fighter and serves really well.
"I am super happy. It seems like German conditions suit me. This is my fifth [ATP Tour] title and it will give me confidence for the upcoming tournaments."
At the Estoril Open, Albert Ramos-Vinolas emerged victorious from a marathon final against Cameron Norrie.
The 33-year-old was a set and a break down but rallied to win 4-6 6-3 7-6 (7-3) in two hours and 44 minutes in Portugal.
It was the Spaniard's third Tour title and took him to a 16-5 record on clay this season. He has at least seven more wins on the surface than any other ATP player this season.
Basilashvili beat Bautista Agut in the championship match in Doha last year, but the Spaniard came out on top on this occasion, winning 6-3 6-4.
The experienced second seed had not won a tournament since he was crowned Qatar Open champion in 2019, before repeating that feat with a straight-sets success.
Bautista Agut came from a break down in both sets to win the 10th ATP title of his career at the expense of the third seed from Georgia.
The 33-year-old world number 16 dropped only one set this week en route to getting his hands on another trophy.
He said: "It was a really tough final. This year we played without wind, which was much better for me.
"It was not easy to recover after a really tough battle yesterday [in a semi-final win over Karen Khachanov], but I think I did a good job today.
"It is my 10th ATP Tour title and I am very happy."
Spanish second seed Bautista Agut had dashed past Andy Murray and Alejandro Davidovich Fokina for the loss of just three games in Doha, but Khachanov proved far more obdurate.
After a slow start, eventually Bautista Agut picked up the pace and emerged a 2-6 6-3 7-5 winner in two hours and 24 minutes of toil, earning a shot at Georgia's Nikoloz Basilashvili in Saturday's final.
Bautista Agut praised Russian Khachanov for making him scrap for the win.
"He was playing very fast," said Bautista Agut, quoted on the ATP website. "He was playing very solidly, he was defending very well and he hit many unbelievable passing shots. I had to play incredibly well tonight. I enjoyed tonight so much."
Third seed Basilashvili is the defending champion this week and made light work of France's Arthur Rinderknech, tying up a 6-4 6-2 victory.
In Marseille, Greek top seed Stefanos Tsitsipas was stunned by Russian qualifier Roman Safiullin in the Open 13 quarter-finals. World number 163 Safiullin had a staggering 13 break points against the under-performing Tsitsipas serve, and although 11 of those were saved, the two that were not decided the match.
Safiullin, for his part, faced one break point and saved it on the way to a 6-4 6-4 victory. He made his presence felt at the ATP Cup in Australia at the start of the year and, although results have been disappointing since, the 24-year-old is doing well again in France this week.
Next for Safiullin will be last week's Rotterdam champion Felix Auger-Aliassime, who saw off Belarusian Ilya Ivashka 6-3 6-4 in the last match of the day. After losing his first eight finals on the ATP Tour, Auger-Aliassime may well be closing in on a second title in eight days.
Saturday's other semi-final will see Russian second seed Andrey Rublev tackle Frenchman Benjamin Bonzi, who reached the last four of a main-tour event for the first time by crushing Aslan Karatsev 6-1 6-3.
Rublev dug deep to edge out French wildcard and former top-10 player Lucas Pouille 6-3 1-6 6-2.
Top seed Berrettini, ranked nine in the world, has not dropped a set all week.
His impressive run has seen him defeat home hopes Andy Murray and Dan Evans, with one more Briton in the shape of Norrie left to see off in his bid for glory.
Berrettini ensured he will be in the final by claiming a 6-4 6-4 triumph over fourth seed Alex de Minaur in the semi-final on Saturday.
The Italian dropped just four of his 36 points on first-serve and sent down eight aces, with De Minaur only able to force one break point in the entire contest, which he did not take.
"[Making the final] was the goal of the week and now I have one more step," said Berrettini.
"It is a great achievement, especially for the history of this tournament. I am really happy because to beat Alex, I had to play my best tennis."
Berrettini has four tour titles to his name, though this would be his first at ATP 500 level or above.
Victory would also represent the biggest win of Norrie's career – he has lost each of his three previous finals, all at ATP 250 level.
Norrie impressively eliminated Denis Shapovalov to reach the showpiece, beating the Canadian 7-5 6-3.
Shapovalov had earlier finished off a 6-3 6-4 quarter-final win over Frances Tiafoe, a match that could not be completed on Friday due to fading light.
But the second seed could not muster up another victory against a fresher Norrie.
At the Halle Open, Andrey Rublev reached his eighth final since the start of 2020, though his first on a grass court.
Rublev dropped his first set of the week but ultimately prevailed with a 6-1 3-6 6-3 semi-final victory over Nikoloz Basilashvili.
The Russian has won his last four finals at ATP 500 level and will seek a fifth on Sunday.
"It's my first final on grass and in Halle," he said. "I think I can play on every surface and I will try my best again.
"I had good opportunities to break Basilashvili in the second set, some quite easy forehands and I stressed a little showing my emotions.
"I then came back and stayed calm, until the last game. But I won."
Rublev will take on unseeded Ugo Humbert, who held his nerve to edge a thriller against Felix Auger-Aliassime, winning 6-4 3-6 7-6 (7-5).
Humbert beat Alexander Zverev earlier in the week and has had to win a deciding set in all four rounds, while the beaten Auger-Aliassime had previously seen off Roger Federer as part of a dramatic event.
Frenchman Humbert won each of his first two career finals, which both took place last year in ATP 250 events.
After a delay of more than four hours because of rain in London, Berrettini overcame Evans 7-6 (7-5) 6-3 to reach his third ATP grass-court semi-final.
The Italian hit 13 aces and won 81 per cent of his first-serve points against Evans to set up a last-four clash with Alex de Minaur.
Berrettini improved his win-loss record to 24-6 for the season and laid out his ambition to walk away from the tournament with the trophy.
"I didn't serve that well, but I was returning well and I just played better in the last few points of the tie-break," he said.
"After that, I felt more confident. The conditions were really tough, windy and cold, so I took time to adapt a little bit. I am pretty happy with my performance.
"The court condition was really good. I expected slippery conditions, but it was like yesterday.
"I came here to win the tournament, that is my goal. Now I am two steps away. I am happy with the way I am playing, and my mental attitude is really good."
Up next for Berrettini is Australian De Minaur, who came from behind to defeat Marin Cilic 3-6 6-3 6-4.
The 22-year-old won 73 per cent (22/30) of his second-serve points and saved six of the seven break points he faced as he moved to 16-12 for the season.
In the battle of the British players, Cameron Norrie beat Jack Draper 6-3 6-3, while Denis Shapovalov was leading Frances Tiafoe 6-3 when their match was suspended due to fading light. They will resume on Saturday.
At the Halle Open, Andrey Rublev reached his sixth ATP Tour semi-final of the year thanks to a 7-6 (7-4) 6-2 win over 2011 champion Philipp Kohlschreiber.
"I am happy with my performance to reach the semi-finals for the first time," Rublev said. "The first set was really tough. He was 3-0 up in the tie-break and I came back, which was the key.
"After the first set, I think he mentally went down and I was pumped up. I hit a couple of good returns in the first game of the second set."
Russian Rublev will face Nikoloz Basilashvili in the last four after the Georgian defeated Lloyd Harris 6-4 7-6 (7-5).
In the day's other quarter-finals, Felix Auger-Aliassime beat Marcos Giron 6-3 6-2 and Ugo Humbert overcame Sebastian Korda 6-2 6-7 (5-7) 6-4.
Carreno Busta demolished seventh seed Soonwoo Kwon 6-4 6-0, while Casper Ruud was stunned in straight sets by wildcard Alcaraz.
The 17-year-old backed up his victory over the veteran Feliciano Lopez in style, becoming the youngest ATP Tour semi-finalist since Alexander Zverev at Hamburg in 2014 with a 6-2 6-4 win.
Albert Ramos-Vinolas will face Carreno Busta after beating Norbert Gombos 6-7 (4-7) 7-5 6-4, while Alcarez is next on the agenda for compatriot Jaume Munar, who was taken the distance by Ilya Ivashka having overcome second seed Fabio Fognini on Thursday.
Taylor Fritz continued his solid form this season with a straight sets victory over Aljaz Bedene to reach the semi-finals of Sardegna Open.
Fritz is playing as the US number one for the first time in Cagliari this week and he made light work of Bedene, prevailing 6-3 6-4 to book a showdown with third seed Lorenzo Sonego, who beat Yannick Hanfmann 3-6 7-6 (8-6) 6-3 despite being 3-5 down in the second.
Seeded second and ranked 30 in the world, Fitz improved his 2021 record to 11-5 and will look to avenge a defeat in his only other meeting with Sonego on clay, when he lost at Roland Garros last year.
Lorenzo Musetti thrilled as he saved four match points to sink top seed Dan Evans on Thursday but bowed out after another epic, losing to Serbia's Laslo Djere 6-4 4-6 6-2 in two hours and 21 minutes.
Djere, who beat sixth seen John Millman en route to the quarter-finals, will face Nikoloz Basilashvili after the battle between seeds four and five ended 7-5 4-6 6-3 in the Georgian's favour.
Cilic, who was also celebrating his 34th birthday, was given a thorough test initially by the wild card, but Thiem ultimately faded away.
After losing the first set, second seed Cilic eventually won 6-7 (6-8) 6-2 6-4 as he succeeded in putting the Austrian's serve under intense pressure, setting up a quarter-final with either fifth seed Botic van de Zandschulp or Liam Broady.
Reacting to the victory, Cilic said: "I think it was a fantastic match. I think it was an amazing, amazing level.
"We don't get so many chances to play [a] second-round [match] in front of this big [of a] crowd, so thank you so much. It was a fantastic birthday."
Maxime Cressy, seeded fourth, was far more comfortable as he defeated fellow American Sebastian Korda 6-4 6-4 thanks to a break of serve in either set.
The only other seed in action was seventh favourite Adrian Mannarino. The Frenchman came unstuck against his countryman Constant Lestienne, who won 7-6 (7-2) 6-4.
At the Sofia Open, two high seeds were surprisingly dumped out on Wednesday.
Home favourite Grigor Dimitrov, seeded third, was beaten in straight sets by Ilya Ivashka, who showed commendable focus to see out a 6-3 7-6 (7-3) win despite the crowd's support of his opponent.
Number six seed Nikoloz Basilashvili was the other big name to suffer a shock defeat, with 38-year-old Fernando Verdasco beating him 7-6 (7-1) 7-5 as the seven-time ATP Tour title winner rolled back the years.
Lorenzo Musetti bucked the trend, however. Seeded fourth, the Italian was made to sweat by Bulgaria's Alexander Lazarov as the first two sets went to tie-breaks, but the Italian had too much in the end as he won 6-7 (5-7) 7-6 (10-8) 6-2.
Earlier in the day, Musetti's compatriot Lorenzo Sonego comfortably dealt with Bernabe Zapata Miralles 6-3 6-4.
The 20-time grand slam champion eased to a 6-3 6-2 win in the Spanish capital, assuring him of a record-extending 369th week at the top of the ATP world rankings and teeing up a mouth-watering contest with one of his greatest rivals.
The Serb hailed his victory as representing his best performance of 2022, saying he was pleased with his progress after falling to a final defeat against Andrey Rublev on home soil in last month's Belgrade Open.
"[It was] probably the best match of the year so far for me," Djokovic told Tennis TV after the win. "I haven't played many matches and am still finding my groove.
"But [it's] a very good win against a very tricky opponent. I had a week, 10 days to get ready for this match, and for this tournament I have done everything I possibly can to build my fitness and also improve on all aspects of the game, on the court.
"I'm really glad that it paid off, because I felt good on the court. It's the right process, and it's the right direction."
Djokovic will face Murray for the 37th time, and the first time since 2017's Qatar Open, after the three-time grand slam winner rolled back the years to overcome Shapovalov 6-1 3-6 6-2 in an absorbing two-hour contest later on Tuesday.
The duo are one of only two male pairs to have met in each of the four grand slam finals (along with Djokovic and Rafael Nadal), while Djokovic holds a 25-11 lead across their previous meetings and has won both of their two encounters at the Madrid Open, one of which was their first-ever meeting in 2006.
Elsewhere in the draw, it was a day of few shocks in Spain as seeds Rublev and Carlos Alcaraz ensured their own progress to the last 16.
Rublev recovered from a poor first-set showing to down Great Britain's Jack Draper 2-6 6-4 7-5, while home favourite Alcaraz remained on course for a potential meeting with compatriot Nadal after earning a 6-3 7-5 victory over Nikoloz Basilashvili.
Finally, 2014 US Open champion Marin Cilic beat Albert Ramos-Vinolas 6-3 3-6 6-4 in a competitive first-round affair to set up Wednesday's enticing last-32 clash with world number three Alexander Zverev.
After 14 months out following knee surgery, Federer was back in action at the Qatar Open on Wednesday, defeating friend and practice partner Dan Evans in three sets in the last 16.
But Nikoloz Basilashvili, Federer's next opponent, proved a step too far in the quarter-finals on Thursday as the 39-year-old Swiss superstar went down 3-6 6-1 7-5.
Federer breezed through the opener and recovered from a tough second set to forge a match point in the decider, but Basilashvili stuck with his "idol" and earned a first career win against the 20-time grand slam champion.
The result would not put a dampener on Federer's week, however.
"I'm already over it," Federer said. "I mean I would have loved to play tomorrow - don't get me wrong, you know - but at the same time I'm also happy to get a rest.
"I'm happy how I played today. I'm happy how I did yesterday. I'm happy I was back on the Tour. I'm pleased I came here to Doha.
"So it's a really, really positive return for me.
"I'm actually happy that I was able to play back-to-back three-set matches against top players. That's an important step forward for me. This is a stepping stone."
But Federer announced later on Thursday he had made the decision to return to training, putting on hold plans for a second tournament of the season.
He had been set to join a number of other big names in Dubai, although Spanish great Rafael Nadal has rejected a wildcard.
Federer wrote on his social media pages: "It's been great to be back on the @atptour, loved every minute playing in Doha once again. A big thank you to the best and loyal team that helped me get here.
"I've decided it's best to go back to training and as a result, I've decided to withdraw from Dubai next week."
Federer is an eight-time Dubai champion, most recently in 2019 when he defeated Stefanos Tsitsipas in the final.
Roger Federer's return to the ATP Tour lasted just two matches at the Qatar Open as he was beaten in the last eight by the brilliant Nikoloz Basilashvili.
Goffin, who won the event in 2014, fell to a 6-3 6-3 defeat to three-time champion Gilles Simon, who is set to retire at the end of the 2022 campaign.
Simon needed 93 minutes to wrap up his victory over the four-time grand slam quarter-finalist, teeing up a round-of-16 meeting with Italy's Lorenzo Sonego.
"We've known each other for a very long time and I’m always happy to play against [Goffin] one more time before the end," Simon said after the win.
"It was really perfect from the start until the end and that's how I won this one."
Elsewhere, Basilashvili was a set down in his meeting with Bergs when the Belgian qualifier was forced to retire through injury, allowing the fifth seed to progress to a meeting with Arthur Rinderknech.
Rinderknech downed Spain's Jaume Munar in three sets to reach the next round, while Emil Ruusuvuori comfortably saw off Jiri Lehecka 6-4 6-4.
Fourth seed Hurkacz, who was beaten in the second round in Melbourne by Adrian Mannarino, saw off wildcard Tsonga 6-4 7-6 (9-7) in one hour and 32 minutes.
"Jo is an amazing player and competitor," Hurkacz said of the former world number five.
"He brought a really high level and it was a really tough match for me and he is definitely on the way back.
"Jo has achieved so much in his career and playing against him you can see why with his shots and his serve. I appreciated playing him and I am happy with the win today."
World number 11 Hurkacz will face either Lorenzo Musetti or Mikael Ymer in the second round.
Sixth seed Cameron Norrie overcame Ugo Humbert 6-2 6-4, while eighth seed Nikoloz Basilashvili was dumped out by American Mackenzie McDonald 6-3 6-2.
McDonald will face Alex de Minaur in the next round after the Australian swatted aside David Goffin 6-0 6-3.
"I think I played really well," De Minaur said. "I played exactly how I wanted to. My coach and I set up a plan about how we wanted to play this match and I managed to execute it.
"It is a very good feeling. Mentally I thought I was very solid and positive and that is one of the main aspects I am trying to focus on this year."
The three-time grand slam winner made his tour return last month after requiring two surgeries on a left foot injury, but rallied after losing his first set against Opelka to secure a memorable 3-6 7-5 6-2 first-round victory.
Speaking after booking a second-round meeting with Laslo Djere, who beat Borna Coric 6-2 6-7 (3-7) 6-2, Wawrinka said his desire to play at the top level never left him, despite his injury woes.
"I think in general I was feeling good on the court, physically I was feeling great," he said. "For sure, when you don't win a match in more than a year, you start to think about it more than you should and not focus on the right things.
"In general, I think it was a great match, a great battle. I stayed positive. I started to feel much better with my tennis by the end of the second set, and in the third set. I'm really happy with this victory.
"After two surgeries, at my age I could easily have stopped playing because my career is way better than what I expected when I was young.
"But I still have this fire in me. I still believe that I can play great tennis. I still believe that I make some big results, maybe not now, but in a few months."
Fellow former grand slam champion Dominic Thiem, who also returned from a long injury-enforced absence last month, fared less well, going down in straight sets to Fabio Fognini.
Having lost to Andy Murray at the Madrid Open last time out, 2020 US Open winner Thiem is still chasing his first victory of the year after losing 6-4 7-6 (7-5) to the 34-year-old Italian.
Elsewhere, 13th seed Denis Shapovalov edged an epic three-hour contest against Italy's Lorenzo Sonego 7-6 (7-5) 3-6 6-3, responding after being handed a game penalty for shouting expletives at a partisan crowd in Rome during the second set.
The frustrated Canadian was heard to tell spectators: "Shut the f*** up" when being booed after complaining about a second serve being called out by the umpire, but eventually regained his composure to claim a three-set win.
Shapovalov will face Georgia's world number 25 Nikoloz Basilashvili in the second round after he overcame Daniel Evans 7-6 (7-2) 6-2, while the only other seed to play on Monday, Spain's Pablo Carreno Busta, cruised to a 6-3 6-2 victory over Federico Delbonis.
World number two Medvedev was making his first appearance since March after undergoing a hernia operation and fell to a 6-2 7-6 (7-5) defeat.
The Russian's rustiness was clear in the last-16 tussle as he racked up seven double faults and struggled to make inroads on Gasquet's second serve, with the Frenchman winning 61 per cent of points behind it.
It was the first time Gasquet overcame an opponent ranked in the top two since beating Roger Federer at the 2005 Monte Carlo Masters.
Next up for Gasquet will be Kamil Majchrzak, who beat Marco Cecchinato 6-2 6-3.
At the last-32 stage, Fabio Fognini went down 6-4 6-3 to Thanasi Kokkinakis and Albert Ramos-Vinolas succumbed to a 7-6 (7-5) 6-4 loss against Christopher O'Connell.
Johan Nikles, Nikoloz Basilashvili and Joao Sousa also advanced to the second round.
Top seed Cameron Norrie cruised into the quarter-finals of the Lyon Open by beating Francisco Cerundolo 6-4 6-4.
The Briton will face another Argentinian next in the form of Sebastian Baez, who came from a set down to beat Oscar Otte 5-7 6-4 6-2.
Alex De Minaur also had to rally for a 1-6 6-3 6-2 win against Ugo Humbert, with Yosuke Watanuki awaiting in the last eight after the world number 263 beat Soonwoo Kwon 6-3 6-4.
World number four Medvedev fell to a 6-4 6-7 (7-9) 6-3 loss against three-time major winner Wawrinka, who will next meet Mikael Ymer after his straight-sets win over home favourite Gregoire Barrere.
Second seed Hurkacz did not experience such struggles against Thiem as the Pole cruised to a 6-3 6-4 as he served 12 aces in a dominant performance in Metz.
Hurkacz will face Arthur Rinderknech in the next round after the Frenchman battled to a 7-6 (13-11) 6-3 triumph over fifth seed Nikoloz Basilashvili.
Alexander Bublik, the seventh favourite in France, was also made to work for his progress to the quarter-finals, downing Emil Ruusuvuori 6-7 (4-7) 6-4 6-3.
Fourth seed Holger Rune awaits Bublik after Benjamin Bonzi retired due to injury when 6-4 4-1 down against the Dane.
Lorenzo Sonego coasted through his second-round clash to defeat Gilles Simon 7-6 (7-2) 6-4 to tee up a last-eight clash with Sebastian Korda.
At the Queen's Club Championships, second seed Matteo Berrettini came from a set down to finally see off Denis Kudla, winning 3-6 7-6 (7-5) 6-4.
Defending champion Berrettini was troubled by the world number 82, and was just a tie-break away from suffering an upset, but the Italian came through as he belted down 22 aces in the match.
Berrettini faces Tommy Paul in the quarter-finals after the American beat Stanislas Wawrinka in straight sets, 6-4 6-1.
Ryan Peniston carried on from knocking out number one seed Casper Ruud by beating Francisco Cerundolo 6-0 4-6 6-4, setting up a last eight clash with Filip Krajinovic after he came from behind to defeat Sam Querrey 4-6 6-3 6-4.
Murray needed less than an hour to record a 6-2 6-0 triumph over the big-serving American, setting up a last-16 meeting with Australia's Max Purcell.
Three-time grand slam champion Murray hobbled out of his last meeting with Querrey at the Wimbledon quarter-finals in 2017, a defeat which marked the start of his injury woes.
But he produced a polished display on the grass to wrap up his eighth career win over the American on Tuesday.
Elsewhere in Rhode Island, eighth seed James Duckworth battled to a 6-2 4-6 6-2 win over Liam Broady, and seventh seed Jiri Vesely raced to a 6-4 6-2 triumph against Spain's Feliciano Lopez.
Meanwhile, Thiem ended a 426-day wait for a Tour-level win by overcoming Finland's Emil Ruusuvuori 3-6 6-1 7-6 (7-5) at the Swedish Open.
The Austrian, who has won 17 career titles, halted a run of 10 consecutive losses on the ATP Tour with his first triumph since he beat Marton Fucsovics at the last-32 stage of the Internazionali d'Italia in May 2021.
The 28-year-old expressed his delight at ending his drought in the aftermath, saying: "It's a long time. My last victory was in Rome in 2021, it feels like a different world somehow.
"Many, many things happened. It was tough, but it was also a very good experience I think, for life in general. I'm so happy that I got this first victory here today."
Thiem will face fourth seed Roberto Bautista Agut in the last 16, while Sebastien Baez will take on Alejandro Davidovich Fokina after the former beat Fabio Fognini in straight sets.
The round of 32 did see two shock exits, however, with seventh seed Holger Rune succumbing to a 6-3 6-3 defeat against Switzerland's Marc-Andrea Huesler, and Nikoloz Basilashvili being forced to retire at one set down against Hugo Gaston.
Also, Francisco Cerundolo beat Pedro Sousa to set up an enticing clash with top seed Casper Ruud.
