Despite breaking Dzumhur's serve at the first opportunity, Auger-Aliassime – who failed to get past the first round of the Australian Open – conspired to lose the first set, as his counterpart broke back straight away and eventually closed out the set in the tie-break.
That seemed to be the shot in the arm Auger-Aliassime needed, as the Canadian world number 21 cruised through the second and third sets – enjoying three breaks of service in the latter, though it still took him two hours, 31 minutes to clinch Tuesday's win.
Victory sets up a second-round clash with France's Pierre-Hugues Herbert, who eliminated Sergiy Stakhovsky 6-2 7-5, while seventh seed Filip Krajinovic also went through in straight sets, beating Enzo Couacaud 6-4 6-4.
Kazakhstan's Alexander Bublik will face second seed David Goffin next after a gruelling 6-7 (7-2) 6-1 7-6 (7-2) win over Henri Laaksonen.
Mikael Ymer and Emil Ruusuvuori also secured passage to the last 16.
James Duckworth was the highest seed in action at the Pune Open on Tuesday and the Australian – who is the sixth favourite – had little difficulty getting past Peter Gojowczyk 7-6 (7-5) 6-4.
The home contingent had mixed success. Arjun Kadhe lost out 6-2 6-4 to Jiri Vesely, but Prajnesh Gunneswaran ousted Yannick Maden 7-6 (7-4) 7-6 (7-5) to progress.
Ilya Ivashka, Nikola Milojevic, Egor Gerasimov and Roberto Marcora all made it through as well, the latter set to face top seed Benoit Paire next.
Bublik is seeded sixth in Montpellier but is coming off a second-round exit at the hands of Gael Monfils in Melbourne.
However, the world number 35 moved into the last eight on Wednesday, though he needed three sets to do so against Pierre-Hugues Herbert, the qualifier taking the second set to force a decider.
Bublik claimed a 6-3 6-7 (2-7) 6-2 win and will face Roberto Bautista Agut in the quarters.
Bautista Agut claimed the first set against Gilles Simon 6-4 and was 3-2 up in the second when the Frenchman retired injured.
Besides Filip Krajinovic's 6-4 7-6 (7-2) defeat of Jo-Wilfried Tsonga, the rest of the day's action comprised of round of 32 matchups. Seventh seed Ugo Humbert crashed out in three sets to four-time champion Richard Gasquet, while Adrian Mannarino and Mackenzie McDonald both prevailed.
Meanwhile, at the Tata Open Maharashtra, Lorenzo Musetti claimed his first win of 2022 following an impressive 2021.
The 19-year-old, who progressed to the last 16 at Roland Garros last year, fought back from losing the first-set tie-break to defeat Aleksandar Vukic in three sets.
"Aleksandar was playing good, he was serving really big, and I tried to stay calm even in the second set, [which] I didn't win. It didn't go the way that I planned. But the third set was a really great set," Musetti said.
"I stayed really focused with the right attitude and I won a great match with a tough opponent. I'm really happy to be here and to be in the quarter-finals."
Also victorious in Pune were Joao Sousa, Daniel Altmaier and Kamil Majchrzak, whom Musetti will face next.
The 24-year-old went into Sunday's match without a career win over a top-five opponent and having lost two finals in a career-best season in 2021.
He was not to be denied this time, however, powering his way to a 6-4 6-3 win over the world number three in Montpellier in just one hour and 10 minutes.
The Kazakhstani lost just four points behind his first serve throughout the match, the only blip coming when he handed Zverev a break back to love after moving 3-2 ahead in the first set.
After regaining the advantage, Bublik survived two more break points before closing out the opener and the pressure only increased on the Zverev serve in the second set.
Bublik broke to love to lead 3-2 and again in the final game as Zverev's resistance wilted.
He can now look forward to rising to a career-high place in the rankings when the updated list is published on Monday.
Gasquet – three times a champion in Montpellier and once in Lyon – needed just 87 minutes to overcome Simon in his first-round match on Wednesday and set up a last-16 tie against Feliciano Lopez, who beat eighth seed Ugo Humbert.
The other first-round match saw Adrian Mannarino defeat Alexei Popyrin and progress to face top seed Gael Monfils.
Two last-16 matches also took place, with world number 23 Grigor Dimitrov, who entered as a wildcard and was seeded fourth, going down 6-7 (4-7) 6-4 7-5 to Gregoire Barrere.
Third seed Denis Shapovalov also suffered a quick exit as he lost to Vasek Pospisil 6-2 6-3.
Meanwhile, at the Pune Open, second seed Ricardas Berankis saved two set points as he rallied to defeat Cedrik-Marcel Stebe 7-6 (7-2) 6-1.
World number 73 Berankis will go up against Yuichi Sugita, who progressed without playing after Viktor Troicki withdrew due to fever, in the quarter-finals. Jiri Vesely and Ilya Ivashka also progressed.
The Belarusian saw off Aljaz Bedene, with a 6-4 7-6 (7-4) win carrying him through to the quarter-finals of the Open Sud de France.
Bedene was British number two to Murray heading into Wimbledon in 2016, when the Scot won his third grand slam title, but has since reverted to representing his native Slovenia.
He put up a fight against Gerasimov, who sank Murray in straight sets on Tuesday, but the 28-year-old from Minsk took three of his five break-point opportunities and held his nerve in the tie-break to advance.
The success for Gerasimov sets up a last-eight clash with Spain's Alejandro Davidovich Fokina, who scored a fine 7-5 6-2 win over Polish fourth seed Hubert Hurkacz.
Top seed and world number 13 Roberto Bautista Agut raced to a 6-0 6-3 victory over French qualifier Gregoire Barrere, but Croatian third seed Dusan Lajovic tumbled out, losing to Dennis Novak in straight sets.
At the Singapore Open, second seed John Millman fell by the wayside, the Australian slumping 6-4 6-1 to compatriot Matthew Ebden.
French top seed Adrian Mannarino encountered no such problems, ending the hopes of Roberto Marcora with a 6-3 7-5 win to join Ebden in the quarter-finals.
The Belgian came from a set down to prevail 5-7 6-4 6-2, securing a fourth consecutive win over Bautista Agut.
It made him just the third non-Frenchman to lift the trophy in Montpellier, following in the footsteps of Tomas Berdych and Alexander Zverev.
The world number five sent down 13 aces and converted four of his seven break points en route to his fifth tournament win.
Alexei Popyrin claimed his maiden ATP Tour title with victory over Alexander Bublik in the final of the Singapore Tennis Open.
Popyrin, who stunned Marin Cilic in the last four, triumphed 4-6 6-0 6-2.
The Australian lost only six points on his serve, wrapping up the win in 84 minutes.
"It feels unbelievable," said Popyrin. "A lot of sacrifice, a lot of dedication and a lot of hard work went into this from myself, my team and my whole entire family.
"I have done it for everybody who was involved. I am just so proud that I could actually get it done."
Having previously lost in two semi-final appearances at the tournament in Montpellier, Goffin made it third time lucky as he came out on top in a deciding set against his opponent from Belarus.
The second seed claimed the crucial break in the seventh game, then quickly consolidated his lead with an impressive hold. Gerasimov forced him to serve out for the win and Goffin duly delivered under pressure, clinching on his second match point.
The world number 15 – who won two of his four career titles in 2017, but none since - will next face Roberto Bautista Agut, who defeated Peter Gojowczyk 7-5 6-1 after just one hour and 16 minutes on court.
At the Singapore Tennis Open, Alexei Popyrin caused an upset as he defeated Marin Cilic via a pair of tie-breaks in a hard-fought semi-final.
Popyrin prevailed 7-6 (7-5) 7-6 (7-2) to knock out Cilic, who had survived two match points before eventually beating Kwon Soon-woo in the previous round.
Alexander Bublik is also through to the final, guaranteeing a first-time winner on the ATP Tour at the tournament.
Bublik made the final of the Antalya Open earlier this year but had to retire due to an ankle injury. He will hope for better fortunes on Sunday, securing another opportunity for a maiden title after battling back from a set down to see off Radu Albot.
World number 10 Goffin has started the season in impressive form and remains on course for a first ATP Tour title since Tokyo in 2017.
The second seed bested Bublik's varied strokeplay over the course of one hour and 47 minutes, with the 22-year-old even serving underarm in a bid to deceive Goffin as a gripping second-set tie break slipped away.
Goffin will face Pierre-Hugues Herbert in the quarter-finals after the home favourite claimed an impressive 7-6 (7-2) 7-5 triumph against Felix Auger-Aliassime.
Herbert was a finalist in Montpellier last year, while countryman Richard Gasquet remained in contention for his fourth win at the event by beating Feliciano Lopez 6-7 (1-7) 6-4 6-2.
Vasek Pospisil, a winner in the all-Canadian clash with Denis Shapovalov on Wednesday, is up next for Gasquet, who retained his place in a draw stacked with French hopes.
Gael Monfils came from a set down to beat compatriot Adrian Mannarino 4-6 6-1 6-4, joining Gregoire Barrere – a shock winner over Grigor Dimitrov a day ago - in the last eight.
Filip Krajinovic will be Barrere's next opponent after demolishing qualifier Mikael Ymer 6-1 6-1, with Norbert Gombos also seeing off Emil Ruusuvuori in straight sets.
At the Pune Open, top seed Benoit Paire was dumped out in straight sets by Roberto Marcora – the Italian qualifier prevailing 6-4 6-4.
Elsewhere, seeds Ricardas Berankis, Kwon Soon-woo, Yuichi Sugita, James Duckworth and Egor Gerasimov emerged unscathed to reach the quarter-finals.
Huesler arrived in Montpellier on a high from victories over German duo Alexander Zverev and Oscar Otte for Switzerland in Davis Cup qualifiers last weekend.
The left-hander saw off 18-year-old Frenchman Van Assche 6-3 3-6 6-3 and will get a chance to improve his perfect record against Rune to 3-0.
Italian Lorenzo Sonego sent eighth seed Benjamin Bonzi packing with a 6-4 6-3 win, while French-born American Maxime Cressy progressed with a 6-3 6-2 defeat of Antoine Bellier.
Arthur Rinderknech and Quentin Halys also advanced on home soil, along with Marton Fucsovics and Filip Krajinovic.
Seeking his third title at the event, world number nine Monfils triumphed 7-6 (7-4) 6-2 in 82 minutes against the Serbian.
Monfils saved all three break points against Krajinovic in an impressive victory that maintains France's perfect record of ensuring a home player reaches the final of the Montpellier tournament.
The 2010 and 2014 champion will meet Pospisil on Sunday after the Canadian defeated David Goffin in three sets.
Goffin served for the match in the decider but lost three decisive games in a row as Pospisil triumphed 6-3 1-6 7-5 in two hours and 21 minutes.
At the Pune Open, Australia's James Duckworth was beaten 7-6 (7-2) 6-4 by Egor Gerasimov in the semi-finals.
The Belarusian dropped only five points behind his first serve as he reached a maiden ATP Tour final, where he will face Jiri Vesely.
Vesely saved match points for the second round in a row, outlasting second seed Ricardas Berankis 6-7 (8-10) 7-6 (7-3) 7-6 (9-7) in a draining encounter to secure a first final in nearly five years.
Bublik's only career title to date came in Montpellier last year, but he will not defend his crown after losing in three sets to Gregoire Barrere in the first round on Wednesday.
That remarkable 6-4 6-7 (12-14) 7-6 (7-3) reverse represented Bublik's seventh loss in seven singles matches so far this season.
The 25-year-old, who has fallen to 50th in the rankings, has lost nine in a row going back to last year.
Bublik worked hard to stay in Wednesday's match in the second set, fending off three match points in the tie-break, but he could not show the same resilience in the decider as Barrere prevailed.
With the champion out, those hoping to take the title this week found life a little easier elsewhere.
Both Jannik Sinner and Alejandro Davidovich Fokina benefited from walkovers, albeit the Spaniard first had to dig in to level his match against Ugo Humbert, who had taken the opener 6-1.
The three-time grand slam champion withdrew from the Australian Open after suffering the injury during last year's Davis Cup Finals.
Murray, 32, was expected to play at the Open Sud de France and Rotterdam Open next month, but announced on Thursday he would not be ready to return.
"The bone bruising is taking longer to heal than first thought," former world number one Murray told UK media.
"So I won't be playing in Montpellier or Rotterdam in February.
"I don't want to rush anything or put a timeline on my recovery, I'm going to listen to my body and step back on the court to compete when the time is right."
After a hip injury threatened to end his career, Murray returned in the second half of last year and won the ATP Tour title in Antwerp in October.
A positive coronavirus test ahead of the Australian Open meant Murray had not played on the main tour ahead of travelling to Montpellier this week.
The three-time major champion, who lost in the final of a Challenger Tour event in Biella earlier this month, revealed on Monday he did not watch the action at the first grand slam of the year "because I wanted to be there myself".
But Murray also suggested practice had gone well in the meantime and he was optimistic of again competing with the world's best.
Tuesday's first-round meeting with Gerasimov brought a bump back down to earth.
The Briton went down 7-6 (10-8) 6-1 to Gerasimov, whose previous outing had been a 6-0 6-1 6-0 humbling at the hands of Melbourne surprise package Aslan Karatsev.
There were no signs of an impending implosion as Murray held his own in the first set, broken in the seventh game but responding instantly to reach a tie-break.
Gerasimov – playing Murray for the first time – converted a fourth set point, though, and his opponent was way off the pace in the second, forced to defend match point just to avoid an embarrassing bagel.
Home hopefuls Lucas Pouille and Gilles Simon were each also eliminated, while eighth seed Jan Lennard Struff went down to compatriot Peter Gojowczyk.
Radu Albot was the only seed in action at the Singapore Open and came through unscathed against John-Patrick Smith, one of three Australian players to depart across Tuesday's four matches at the tournament.
Eighth seed Goffin, ranked at world number 52, prevailed 2-6 7-5 6-1 in Montpellier against the Frenchman.
It looked bleak for Goffin when Bonzi raced into a 3-0 lead in the second set, yet the Belgian reeled off three successive games to get himself back in the fight before snatching a crucial break to make it 6-5, and he did not look back from there.
"During the match I stayed positive even though I was one set and one break down," Goffin said.
"I tried to stay focussed, to stay as long as possible on the court to try to get some confidence, to work on certain things: to be more aggressive, [hit] a better quality of ball.
"At the end it was better and better, so I'm really happy the way I played at the end of the second and in the third."
Next up for Goffin is a tie with Spain's Alejandro Davidovich Fokina, or another Frenchman in the form of Adrian Mannarino, who lost to eventual Australian Open winner Rafael Nadal in Melbourne last month.
Monday's other match saw sixth seed Alexander Bublik defeat Tallon Griekspoor 7-6 (7-4) 7-6 (7-5).
Yuki Bhambri was the story at the Tata Open Maharashtra, as the Indian claimed a first tour-level match win in almost four years, beating Jozef Kovalik 6-7 (10-12) 6-2 7-5.
Bhambri, ranked at 863 in the world, had only played two singles matches on the ATP Tour since 2019 due to a knee injury and had last beat Mirza Basic in Miami in 2018.
"It is a privilege to play at home and it is an important win as well playing at my home tournament," he said. "Each match is tough at tour-level so I am just taking it one match at a time."
Radu Albot beat Federico Gaio in straight sets, while seventh seed Ricardas Berankis was shocked by Quentin Halys and Daniel Altmaier also progressed.
Cressy defeated top seed Holger Rune in the semi-finals but could not repeat that level of performance against Sinner, who is set to rise to 14th in the ATP rankings.
Sinner is the first Italian champion in the history of the tournament, having failed to win a match in his previous two appearances in Montpellier.
He has now collected seven Tour-level titles, with six of those coming on hard courts.
"To be honest, the first key was to hold the serve. In the tie-break I knew he may miss something," said Sinner, who saved all three break points he offered to his American opponent.
"I tried to keep my serve, which I have done, so I am very happy about the first set.
"The second set was a little bit different. I had a couple of good chances but he then served very well.
"I am very happy I broke him here and then I served very well at the end of match."
Sinner claimed two early mini breaks in the first-set tie-break to nose into a 4-0 lead, holding his nerve to win the set at the first time of asking.
A crucial break in the second came in the eighth game, with Sinner clinching victory when Cressy sent a weak volley into the net.
Tsonga last played in the first round in Melbourne last January, when he retired from his clash with Alexei Popyrin, owing to a back injury.
And the former Australian Open finalist was unable to make it a winning return at a tournament where he lifted the trophy in 2019.
Korda eased to a 6-4 6-2 win, with the only home success of the day coming from Ugo Humbert while Lorenzo Sonego saw off Hugo Gaston.
Popyrin, who stunned David Goffin in the first round at this year's Australian Open, was a winner over Christopher Eubanks at the Singapore Tennis Open.
Lloyd Harris beat Popyrin in the second round in Melbourne, but they will not meet again in this tournament after the South African was undone by wild card Adrian Andreev.
Yoshihito Nishioka, Maxime Cressy and Taro Daniel also prevailed.
Last year's Paris Masters champion Rune, who is in the hunt for his first title of the season, beat the Frenchman 7-6 (7-2) 7-6 (7-5) and is yet to drop a set in the tournament.
The 19-year-old looks sharp, but so too does Jannik Sinner. Two years Rune's senior, the Italian made light work of his good friend and compatriot Lorenzo Sonego on Friday.
Sinner took just over 90 minutes to triumph 6-4 6-2 and tee up a semi-final with history-making 18-year-old Arthur Fils.
Fils, the world number 163, beat Quentin Halys to become the youngest Frenchman to reach a tour-level semi-final since Richard Gasquet in 2004.
"I have seen him [Fils] play here at this tournament," Sinner said of Fils, as quoted by the ATP Tour's official website.
"He is very strong physically already. He understands the game very well. He started off the year very well, so it is going to be an interesting match for sure."
Sinner will be the favourite despite Fils' hot streak, while Rune will go up against Maxime Cressy in a rematch of their Australian Open second-round match.
Cressy went down 7-5 6-4 6-4 in that meeting in Melbourne last month, though has the chance for revenge after seeing off Borna Coric 7-6 (7-2) 6-4.
The German world number three was bounced out at Melbourne Park by Denis Shapovalov in straight sets, but he navigated his way safely past Mackenzie McDonald to reach the quarter-finals in Montpellier.
A largely dominant Zverev won 6-2 7-6 (7-5) to set up a clash with Adrian Mannarino in the last eight, and the 24-year-old highlighted how important he felt it was to quickly get back in the swing of things after disappointment in Melbourne.
"I'm going to give myself the best chance to do well in this tournament. But obviously, I'm happy to be back here," he said of the tournament he won in 2017.
"This is a place I really enjoy coming back [to]. After the Australian Open that I had, I wanted to play tournaments before my scheduled tournaments in Acapulco, Indian Wells and Miami. I said Montpellier is the perfect fit for that, and I'm happy to be here."
The same cannot be said for Gael Monfils, who suffered a crushing 6-1 6-2 defeat to Mikael Ymer on home soil.
The Swede was practically flawless throughout, winning a whopping 83 per cent of points on first serve and 89 per cent on second, with Monfils never even sniffing out a single opportunity to break back.
Richard Gasquet is a force to be reckoned with in Montpellier and reached another quarter-final, seeing off South Korean's Kwon Soon-woo 7-5 6-4.
Frenchman Gasquet reached six consecutive finals in Montpellier while in his prime years, being crowned champion in 2013, 2015 and 2016, and finishing runner-up in 2014, 2017 and 2018. Now 35, his last quarter-final appearance in an ATP main tour event came in August at Winston-Salem.
Gasquet's fellow Frenchman Mannarino sank the hopes of defending champion David Goffin, scoring a sparkling 6-4 6-2 victory.
Fourth seed Nikoloz Basilashvili, who lost twice to Andy Murray in Australia last month, this time fell to world number 152 Damir Dzumhur of Bosnia-Herzegovina, sliding to a 6-1 5-7 6-2 defeat. Basilashvili, the world number 21, now has a record of played five, lost five, for the 2022 season.
At the Tata Open Maharashtra in Pune, there was a surprise defeat for Russian top seed Aslan Karatsev. Karatsev won a title in Sydney last month before losing to Mannarino in the third round of the Australian Open, and in his first match since that disappointment he slumped 6-2 7-6 (7-3) to Swedish qualifier Elias Ymer.
Three seeds joined Ymer in the quarter-finals, with Jiri Vesely, Emil Ruusuvuori and Stefano Travaglia all making smooth progress without dropping a set. Vesely faces Ruusuvuori on Friday, while Travaglia awaits Ymer.
The world number three needed just 50 minutes to swat aside Mannarino 6-1 6-0 to set up a last-four match with Mikael Ymer.
Champion at this event in 2017, Zverev's record in Montpellier moved to 9-1 and he has now won seven matches in a row against Mannarino.
Ymer, the world number 83, squandered one match point in his contest with Richard Gasquet but eventually came through 7-5 6-7 (6-8) 6-1.
The best entertainment came in the earlier match between Alexander Bublik and Roberto Bautista Agut, sixth seed Bublik eventually coming through 6-4 2-6 7-6 (7-4) after just over two hours on court.
He will face Filip Krajinovic for a place in the final, the Serbian having beaten Damir Dzumhur in straight sets.
At the Tata Open Maharashtra, rising Italian star Lorenzo Musetti was surprised by world number 95 Kamil Majchrzak.
The Pole, who defeated Musetti's compatriot Andreas Seppi at the Australian Open in straight sets, won 6-2 6-7 (5-7) 6-4 to set up a semi-final against Emil Ruusuvuori.
The other semi-final sees Joao Sousa, into the last four at Tour level for the first time since St Petersburg in 2019, face Elias Ymer.
"I think I played a great match," said Sousa after beating Daniel Altmaier in straight sets. "I was very solid. I knew it was going to be a tough match, so I was prepared to run a lot."
