Number six seed Paire was a 6-3 7-6 (7-2) winner in his first-round match against Jozef Kovalik.
The Frenchman was not broken in the match, saving all eight of the break-point opportunities he offered up to Kovalik during a competitive 85-minute contest.
Paire's win follows a quarter-final appearance at the Hamburg European Open last week as the 32-year-old begins to recover from a dreadful first half of 2021.
Prior to his run in Hamburg, Paire had only won two ATP Tour matches this year.
Mikael Ymer had to come from behind before he progressed with a 4-6 6-3 6-3 win over lucky loser Enzo Couacaud.
The top four seeds at the ATP 250 event – led by Wimbledon semi-finalist Denis Shapovalov – get a bye through the first round and are therefore yet to begin their campaigns.
At the Croatia Open Umag, meanwhile, Radu Albot came out on top as he claimed a three-set victory over Holger Rune, with top seed Albert Ramos-Vinolas and defending champion Dusan Lajovic not in action on day one.
Lithuania's Berankis forced Paire to a first-set tiebreak but, due to rolling his ankle earlier in the game, then had no option but to retire hurt.
His retirement granted Paire just his third ATP win of 2021.
At the same event, there was also a first ever ATP win for Sebastian Baez, who coasted past Corentin Moutet 6-1 6-2 to claim a straight-sets victory.
In the Nordea Open, held in Bastad, Italian Cecchinato triumphed over former world number seven Gasquet 6-2 5-7 6-3 to reach the second round.
The 2018 Roland Garros semi-finalist, who fought off Gasquet's second-set comeback, will now face either Federico Coria or Francisco Cerundolo in the next round.
In the other first-round ties, Yannick Hanfmann recorded a straight-sets win over Thiago Monteiro, while Elias Ymer, playing in his home country, beat Taro Daniel 6-2 6-7 (6-8) 6-4.
Paire was drawn against compatriot Gregoire Barrere in a battle of two home hopefuls, but the more established player triumphed in straight sets.
The 20th-ranked star broke in the third game of the opener and protected his advantage in straightforward fashion.
There were no breaks in the second, but Paire dominated the tie-break to claim a 6-4 7-6 (7-1) victory.
Barrere was not the only Frenchman to exit the competition, as former world number seven Richard Gasquet went down 6-3 3-6 7-5 to Mikael Ymer.
Ymer will now meet Stefanos Tsitsipas, the defending champion and number two seed, who was granted a first-round bye.
In the final match of the day, Alexander Bublik defeated Marton Fucsovics 6-4 7-5.
The Frenchman - ranked no. 49 in the world - marked his debut at the ATP 250 event by ending a three-match losing run, progressing after one hour and 37 minutes on court.
Paire, who recorded five breaks of serve en route to victory, will be up against either James Duckworth or Emil Ruusuvuori next.
"[I am] happy about the win," Paire said in his on-court interview. "It was not easy. Alejandro is a very good player, so I had to be focused on my game.
"I had to serve well and do my best and that is what I did and I am very happy and proud of my game today.
"I played well in the first game, it was good for my confidence. The most important thing is to stay on my game and stay focused. I was solid today so I am very happy."
Also in Monday's action, Adrian Mannarino was another seed to suffer an early exit, going down in straight sets to Gianluca Mager.
Miomir Kecmanovic came through a tight two-set contest to knock out Adrian Andreev, including taking the opener via a tie-break.
The Russian won 7-6 (7-5) 6-7 (7-9) 6-4 in a three-hour test, putting him one step away from a second title in 2020.
After triumphing at the Qatar Open last weekend, Rublev, 22, has brought his best form to Australia and had a marginal edge over his 19-year-old Canadian opponent.
At one point Rublev looked like getting the job done in straight sets, but teenager Auger-Aliassime dug deep to force a decider as he sensed the opportunity to grab a first ATP Tour title this weekend.
It went Rublev's way eventually as he advanced to a title match against South African Lloyd Harris, who had earlier beaten American Tommy Paul 6-4 6-7 (3-7) 6-3 in a battle between two qualifiers.
Harris and Paul, both aged 22 and ranked respectively 91st and 90th in the world, were unlikely semi-finalists, and Harris will go into the final as the obvious underdog.
The Auckland Open will crown its first French champion after Benoit Paire and Ugo Humbert won through to its final.
Dating back to the 1950s, the tournament crowned a host of the world's elite players in its early years, including Rod Laver, Roy Emerson and Bjorn Borg, and its list of winners reflects the cosmopolitan nature of the tour.
Yet French success has been in short supply, until now.
Humbert was a 7-6 (7-5) 6-4 winner against American John Isner, sending the 21-year-old left-hander through to his first tour final.
Paire fended off Poland's Hubert Hurkacz 6-4 6-7 (1-7) 6-2 and will target his fourth ATP title.
"I am very happy to play against a compatriot tomorrow," said 30-year-old Paire, quoted on the ATP website. "He is a very nice guy and we will see, but I am very happy to be in the final. It was the first time for me in the [Auckland] semi-finals and now the first time for me in the final."
A day after first and third seeds Fabio Fognini and Karen Khachanov were knocked out, second seed Shapovalov was beaten 7-5 6-4.
Humbert, who broke serve in the final game of each set, described the win as "one of the biggest of my career" as he set up a semi-final against John Isner.
The American was taken to two tie-breaks by Kyle Edmund but prevailed 7-6 (7-5) 7-6 (7-5), his 25 aces getting him over the line as he seeks a third title in Auckland.
Benoit Paire reached his first semi-final at the event, coming from a set down to beat John Millman 3-6 6-1 6-4.
The fifth seed will face Hubert Hurkacz, who spurned six second-set match points in a mammoth tie-break before eventually seeing off Feliciano Lopez 6-4 6-7 (11-13) 6-4.
At the Adelaide International, home favourite Alex Bolt was soundly beaten by Felix Auger-Aliassime, the Canadian needing just 55 minutes to win 6-3 6-0.
He will meet Andrey Rublev in the last four after the Russian battled past Dan Evans 6-4 3-6 6-3 in just over two hours.
Spanish hopes were extinguished in the quarter-finals, where Tommy Paul surprised Albert Ramos-Vinolas to win 6-3 6-4 and Lloyd Harris beat fourth seed Pablo Carreno Busta 6-3 6-3.
Sixth seed Sonego was the headline act on Monday in Stuttgart and made light work of Benoit Paire to win 7-5 6-2.
The Italian will next face Jan-Lennard Struff after the German edged out world number 53 Marcos Giron 7-5 5-7 7-6 (10-8).
"It's my first time here, it's my first tournament on grass and I'm really happy [with] this match," Sonego said after defeating Paire.
"It's never easy to pass from clay to grass, but I enjoy it because if I serve good it is easier for me."
Struff's compatriot Oscar Otte was another first-round winner with straight-sets victory over Daniel Altmaier, while Frenchman Benjamin Bonzi breezed past Feliciano Lopez.
In Rosmalen, poor weather wreaked havoc with the schedule but France's Gaston had little trouble against South Korean Kwon Soon-woo, winning 6-2 6-1 to reach the last 16, and Andreas Seppi battled past Gijs Brouwer in the second round of qualifying.
Seppi will be hoping to make it to the main draw, where world number two Daniil Medvedev is the top seed, with Felix Auger-Aliassime and Taylor Fritz also competing.
Playing in his first match since a fourth-round loss at the US Open, the world number 49 eased past the man ranked one spot below him, reaching the second round at the event for the first time since 2016.
Tiafoe came up with the big shots when he needed them, converting four of five chances to break Paire's serve, as the veteran suffered 30 unforced errors to just 20 winners.
"I definitely took a long time off after the Open, the longest time I’ve taken in the middle of the season in a while, so [I was] a little rusty in the beginning and in the days leading up I wasn’t feeling great," Tiafoe said in his on-court interview.
"Happy to just get a first round under my belt. Obviously tonight was up and down, Benoit wasn’t having the best of days, but you take that."
Tiafoe will face his 32nd-seeded countryman Sebastian Korda next.
NISHIKORI GOOD AS EVER IN DECIDER
Kei Nishikori is among the most reliable players on the ATP Tour in deciding sets, and he prevailed in the third yet again with a 6-7 (5-7) 6-3 6-2 defeat of qualifier Joao Sousa.
Only world number one Novak Djokovic has a better winning percentage than Nishikori's 147-55 mark (72.8) in deciders, and the Japanese veteran put away Sousa with relative ease after dropping the first-set tie-breaker.
Nishikori hit 40 winners with 23 unforced errors in the match, to Sousa's 22 winners and 20 unforced errors. He moves on to face 18th seed Daniel Evans in the second round.
PAUL SPOILS LOPEZ MILESTONE
Feliciano Lopez achieved a milestone just by stepping on the court as he set a record by appearing in his 139th ATP Masters 1000 event – one more than Roger Federer – but that was the only positive as the 40-year-old Spaniard fell to Tommy Paul 6-3 7-6 (7-3).
The 24-year-old American, ranked 60th in the world, won 70 per cent of his service points to prevail in his Indian Wells debut.
Another tournament newcomer, Germany's Daniel Altmaier, defeated former quarter-finalist Sam Querrey 6-2 6-4 in just over an hour.
Another USA veteran, Steve Johnson lost 6-7 (5-7) 6-3 7-5 to Alejandro Davidovich Fokina, who also is playing the event for the first time.
Tsitsipas claimed his first title of the season by retaining his Open 13 Marseille on Sunday and saw off Carreno Busta 7-6 (7-1) 6-1 two days later.
The second seed put the Spaniard under pressure with rasping returns and was broken just the once to set up a meeting with Alexander Bublik.
Paire saved four match points in a 2-6 7-5 7-6 (7-1) win over 2014 US Open champion Cilic and will face compatriot Richard Gasquet in round two.
Cilic knocked the Frenchman out of the Australian Open in the second round last month, but fell at the first hurdle in the United Arab Emirates.
Paire had lost five consecutive matches against the powerful Croatian, including a five-set battle at Melbourne Park, yet he fended off two match points in the second set and as many in third before bossing a decisive tie-break.
World number 37 Dan Evans claimed the scalp of fourth seed Fabio Fognini, winning 3-6 6-4 7-5 on day two of the tournament.
Fifth seed Roberto Bautista Agut was another to suffer a surpise exit as he went down 7-6 (7-2) 7-5 to Jan-Lennard Struff.
Yasutaka Uchiyama, Filip Krajinovic, Pierre-Hugues Herbert and Nikoloz Basilashvili were also first-round winners.
The event in France, created by Serena Williams' coach Patrick Mouratoglou, is aimed at attracting new fans to the sport and is one of the first tennis tournaments to take place since the coronavirus pandemic struck Europe.
The event is played in a league format, with each match consisting of four quarters and a sudden-death fifth if the scores are level.
On Sunday, world number six Tsitsipas defeated Paire 3-1, hitting more than 30 winners en route to victory.
Gasquet beat David Goffin 3-2 after sudden death, with Feliciano Lopez overcoming Lucas Pouille by the same scoreline.
The UTS' first match was won by Alexei Popyrin against Frenchman Elliot Benchetrit, while Matteo Berrettini also claimed a 3-1 victory over Dustin Brown.
Paire was removed from the men's singles draw after the United States Tennis Association (USTA) on Sunday confirmed an unnamed player had returned a positive COVID-19 test.
As the grand slam got under way behind closed doors at Flushing Meadows on Monday, it was reported a number of French players had to abide by strict quarantine rules because they were in contact with compatriot Paire.
Mladenovic confirmed she must adhere to tighter regulations in New York after she beat Hailey Baptiste 7-5 6-2 on day one of the tournament.
Asked if she had been in contact with Paire, she said: "I did a one-hour training session with Benoît a few days ago, but we were on either side of the net, and that is not taken into account.
"On the other hand, I also spent between half an hour and three quarters of an hour, in the hotel lobby, playing cards with different people. Benoit was there.
"It was the fact of having passed that moment, even though we were all masked, that led to the continuation. It was hard for me to accept the situation because I am not part of his close entourage. And I'm sorry and sad for Benoit."
Quizzed how her situation has changed, Mladenovic replied: "I have the right to play my match, but I literally have the right to do nothing else.
"I am not allowed to go to the gym, not to do anything that is public inside the bubble. I can only be completely alone with my brother, who coaches me. It's really a bubble within the bubble."
Mladenovic revealed she had returned two negative COVID-19 tests since it was revealed Paire had tested positive.
It was reported on Sunday that world number 22 Benoit Paire had returned a positive COVID-19 test at Flushing Meadows.
The United States Tennis Association (USTA) confirmed one player would miss the grand slam after contracting the virus but did not reveal their identity.
"A player has tested positive for COVID-19 at the US Open and has been withdrawn from the tournament. The player is asymptomatic," a USTA statement said.
"The USTA, together with its medical advisors and infectious disease specialist from the Mount Sinai Health System, confirmed a positive test result for a player.
"In accordance with New York State Department of Health requirements, and in alignment with CDC guidelines and the tournament health and safety protocols, the player has been advised that they must isolate for at least 10 days.
"In addition, contact tracing has been initiated to determine if anyone must quarantine for 14 days."
The vast majority of players are staying in a bubble at the famous New York venue for the first major since the tennis season ground to a halt in March due to the pandemic.
Mladenovic had to enter what she described as "a bubble within the bubble" at Flushing Meadows after practicing and playing cards with compatriot men's star Paire, who tested positive for coronavirus.
The 27-year-old looked to have adjusted well to the strange setting as she beat Hailey Baptiste in the first round and then led Varvara Gracheva 6-1 5-1 in the second.
But Mladenovic was remarkably beaten 1-6 7-6 (7-2) 6-0, losing her way completely in the decider after allowing Gracheva to recover in the second set.
The 2015 US Open quarter-finalist pulled no punches after the match as she said she lost "partly because of the conditions" and revealed her regret at entering the tournament due to the way she felt she had been treated due to her association with Paire.
"The third set, I can't describe how I feel. It's distress, really," Mladenovic said. "We should sit down and make a list of what we are going through.
"It's not even acceptable what they do. The third set is a total collapse. I'm going to stop there, I don't know what to tell you.
"How can I tell you [the details]? Even to take a step to the right, they have to ask permission. We have no freedom of movement, no identity. I have the impression that we are prisoners, criminals.
"For any movement, they have to ask if we have the right to do it, even though we are tested every day and have had 30 negative tests. It's abominable, the conditions are atrocious.
"If I had known that playing cards for 40 minutes, with a mask, with a player who tested positive and then finally negative, would have these consequences, I would never have played this tournament."
The world number 10, who has recently been praised by both Roger Federer and Rafael Nadal, took the first set 6-2 with ease before Kopriva, making his ATP Tour debut this week, emphatically fought back 6-3 6-2 to secure a memorable comeback victory.
Number three seed Casper Ruud battled past Dennis Novak 6-4 7-6 (7-5), while Benoit Paire, who only had two ATP wins in 2021 before the Hamburg European Open last week, was 6-4 to the good before Tallon Griekspoor was forced to retire due to injury.
Ruud, who has already won two ATP events in 2021 and three in his career, will now meet sixth seed Paire in the quarter-final stage.
But eighth seed Feliciano Lopez did not enjoy similar success as the world number 96 Mikael Ymer fought past the Spaniard to win 6-7 (6-8) 7-6 (9-7) 6-4 and set-up a quarter-final clash with Kopriva.
At the Umag Open in Croatia, third seed Filip Krajinovic breezed past Radu Albot to record a 6-4 6-2 win.
Krajinovic's countryman and number two ranked player Dusan Lajovic also went through with a 7-5 6-4 win against Bernabe Zapata Miralles.
