Belarus proved no match for Australia, who got the job done across their two singles matches on the day in the Czech Republic.
Storm Sanders got the ball rolling as she comfortably saw off Yuliya Hatouka 6-3 6-3 in the first contest of the day in Group B.
Ajla Tomljanovic was made to work a little harder by Aliaksandra Sasnovich as she lost the first set but eventually rallied to overcome her opponent 4-6 6-2 6-3.
It meant the two countries' doubles meeting was a dead rubber – Belarus at least took the opportunity to avoid a 3-0 annihilation as Sasnovich teamed up with Lidziya Marozava in the 6-4 6-4 defeat of Olivia Gadecki and Ellen Perez.
In Group D, Switzerland and the Czech Republic claimed one victory apiece in the singles – Marketa Vondrousova had too much for Viktorija Golubic in a 6-4 6-2 success that gave the Czechs the upper hand.
But Belinda Bencic levelled things up despite the unenviable task of facing world number three Barbora Krejcikova, emerging with a 7-6 (7-2) 6-4 success.
Bencic was then involved in the doubles decider as well, getting the better of Lucie Hradecka and Katerina Siniakova with Jil Belen Teichmann, their 6-3 6-3 win sealing the semis spot for the Swiss.
The Russian Tennis Federation – who eliminated defending champions France on Wednesday – and the United States are due to contest the other semi-final.
Buffalo Bills safety Hamlin went into cardiac arrest during the game against the Cincinnati Bengals on January 2 and spent over a week in hospital, much of that time in critical condition.
Pegula's parents own the NFL franchise as well as the Buffalo Sabres NHL team, and during her second-round win over Aliaksandra Sasnovich in Melbourne, was seen with a number three on her skirt, which is Hamlin's number and became a symbol of appreciation from well-wishers during his recovery.
"I definitely wanted to do something," Pegula said after her 6-2 7-6 (7-5) victory over Sasnovich. "We were kind of figuring out what the Bills and the Sabres were doing, just as far as what was the message.
"I knew they would probably do something and what message were they trying to send. It ended up being [that] the three was the symbol.
“I just thought it would be cool to put on my outfit here. I thought it would be a fun way to kind of connect with the team and then also just show my support."
There had been speculation that the number was related to her ranking, with the 28-year-old coincidentally the WTA world number three heading into the Australian Open, where she is the third seed, but Pegula laughed off the suggestion.
"I saw someone tweet that: 'Why would you put your ranking on your skirt?'. I'm, like, 'No, that’s not why,'" she said with amusement.
Pegula will play the winner of Olivia Gadecki and Marta Kostyuk in round three at Melbourne Park.
World number one and heavy favourite Barty prevailed 6-3 6-2 against Rybakina to claim the Adelaide title for a second time in three seasons.
The 25-year-old recovered from 40-15 down in the seventh game and went on to immediately earn the only break of the opening set.
Barty did not look back as she held throughout the second set, meaning she went 35 consecutive games without losing serve to conclude the tournament.
After securing a 14th WTA singles title in a little over an hour, Barty now turns focus to next week's Sydney Tennis Classic ahead of beginning her Australian Open campaign.
"I feel good leading up to an Australian Open like I have every year," Barty, who hit 17 winners to 13 unforced errors, said after seeing off Rybakina.
"Each and every preparation is unique. We take it for what comes and what it is, move on, try and do the best that we can in every opportunity.
"It has absolutely no effect on the way that I prepare or the way I'm thinking leading forward just because it's a Grand Slam. It doesn't change for us."
Halep, one of Barty's likely rivals for the Australian Open crown, also made a bright start to the year by overcoming Veronika Kudermetova in the Melbourne Summer Set 1 final.
The number two seed, who is aiming to put last year's injury-ravaged and trophyless season behind her, came out on top 6-2 6-3.
She recovered from an early break down in the first set and again in the second to make it 23 career titles, and a first since the 2020 Italian Open.
It could easily have been a different story, however, as Kudermetova had three break points to move 3-0 ahead in the second set, but the Russian could not take full advantage.
Amanda Anisimova defeated Aliaksandra Sasnovich at the Melbourne Summer Set 2 final to claim the first WTA title of the season, and her second overall.
Belarusian qualifier Sasnovich hit back in style to take the second set 6-1 after losing the opener 7-5, but Anisimova recovered from a break down to edge the deciding set 6-4.
In doing so, Anisimova became the first American to win a title on Australian soil since Sofia Kenin's Australian Open triumph in 2020.
The world number 30 came out on top 6-2 6-2 in a little under an hour to set up a meeting with Arianne Hartono, who earlier beat Anna-Lena Friedsam 7-6 (7-5) 3-6 7-6 (7-4).
Fifth seed Marketa Vondrousova also shuffled through to the next round thanks to a 6-2 6-3 triumph against Alison Van Uytvanck, while Zarina Diyas will face tournament favourite and Olympic champion Belinda Bencic next after beating Anna Karolina Schmiedlova in three sets.
Aliaksandra Sasnovich was made to work harder for her 7-5 7-6 (8-6) win against Lesia Tsurenko, with that the longest straight-sets match of 2021 so far at two hours and 30 minutes, according to the WTA.
Greet Minnen and Mandy Minella were also victorious on Tuesday, overcoming Nuria Parrizas-Diaz and Varvara Gracheva respectively.
At Zavarovalnica Sava Portoroz in Slovenia, meanwhile, home favourite Kaja Juvan eliminated top seed Petra Martic with a 6-3 6-4 win. Sweden's Rebecca Peterson was another seed to fall, going down in straight sets to Lucia Bronzetti.
Three-time grand slam winner Kerber headed to Paris as the 21st seed but in good form after victory at the Internationaux de Strasbourg last week.
Kerber, whose last major title came at Wimbledon in 2018, made it seven straight clay-court wins for the first time in her professional career after defeating Elsa Jacquemot on Wednesday at Roland Garros.
However, Sasnovich – who defeated US Open winner Emma Raducanu in the previous round – proved a step too far for 21st seed Kerber, who fell to a 6-4 7-6 (7-5) loss on Court Simonne-Mathieu.
World number 47 Sasnovich next faces Italy's Martina Trevisan, whose best result at a grand slam was the quarter-finals at this competition two years ago.
Trevisan became the first Italian female player to win eight or more matches in a row since Francesca Schiavone in 2017 by defeating Daria Saville 6-3 6-4 in the third round.
Meanwhile, American teenager Coco Gauff negotiated past Kaia Kanepi 6-3 6-4 to tee up a fourth-round clash with 31st seed Elise Mertens, who was a 6-2 6-3 winner over Varvara Gracheva.
Jabeur took one hour and 31 minutes to overcome her stubborn opponent in the German capital, with the world number four triumphing 6-2 7-6 (10-8) to reach her eighth quarter-final of 2022 – each of which have been at WTA 500 level or higher.
The Tunisian will face Sasnovich for a semi-final spot next time out, with the Belarusian having won 26 matches so far this year (including qualifying draws). Only Iga Swiatek (42) and Beatriz Haddad Maia (29) have more victories so far this season on the WTA Tour.
The other seeds in action in Berlin on Thursday also progressed, with Coco Gauff overcoming Wang Xinyu 6-0 6-4 to tee up a clash with Karolina Pliskova, and Belinda Bencic downing Anna Kalinskaya 6-4 1-6 6-1.
Defending champion Ludmilla Samsonova, however, suffered a 6-3 6-7 (5-7) 6-1 defeat to Veronika Kudermetova.
Meanwhile, the Birmingham Classic's top seed Jelena Ostapenko fell to a surprise 3-6 7-5 7-5 last-16 reverse against Ukraine's Dayana Yastremska, who reached her first ever grass-court quarter-final on the WTA tour.
The world number 79 will face Zhang Shuai for a spot in the final four after she fought her way to a 6-2 2-6 7-5 win over Elena-Gabriela Ruse.
Third seed Camila Giorgi will join them in the last eight after recovering from losing her first set against American Lauren Davis to secure a 3-6 7-5 6-2 win.
Kasatkina, who beat Shelby Rogers in the final of the Silicon Valley Classic a couple of weeks ago, dominated Parry from the jump with her powerful serve.
Parry did not stand a chance when Kasatkina would land her first serves fair, with the Russian converting 88 per cent (15-of-17) of those opportunities in the opening frame, before winning 92 per cent (12-of-13) of her total service points in the second set.
Kasatkina, who has now collected three straight-sets wins in a row, will face Australia's Daria Saville in the final after her semi-final opponent, Marta Kostyuk, withdrew due to injury.
Meanwhile, at Tennis in the Land, Liudmila Samsonova continued her red-hot form with a 6-1 6-2 semi-final victory over Bernarda Pera to book her place in the final.
Samsonova, who is yet to drop a set at the tournament, dominated both with her serve and in her return game, winning 89 per cent (32-of-36) of her service points and 56 per cent (24-of-43) of her return points. She created nine break point opportunities, while allowing zero.
She will play Aliaksandra Sasnovich in the final after the Belarusian outlasted France's Alize Cornet 6-7 (5-7) 7-5 6-3 in a match that lasted two hours and 53 minutes.
Sasnovich shot herself in the foot early with four double faults in the opening set, but cleaned it up as the match wore on, committing just two the rest of the way.
World number two Kontaveit recorded a 6-3 6-2 victory over Peterson on the clay in Germany, setting up a last-eight meeting with eighth seed Andrea Petkovic.
The Estonian's win was her third over Peterson and maintained her 100 per cent career record against Swedish opponents, taking her to seven such successes.
Kontaveit was joined in the quarter-finals by fellow seeds Aliaksandra Sasnovich, who beat Aleksandra Krunic 6-3 6-2, and Maryna Zanevska, who downed Alexandra Cadantu-Ignatik 6-4 6-1.
They will face off for a semi-final spot on Thursday, while Anastasia Potapova will meet 2021 French Open champion Barbora Krejcikova after beating Maria Carle in straight sets.
At the Palermo Ladies Open, meanwhile, third seed Zhang withdrew from her last-16 clash with home favourite Jasmine Paolini before taking to the court, citing an unspecified injury.
Fellow seeds Sara Sorribes Tormo and Anna Bondar both reached the quarter-finals with straight-sets wins, however, with the pair set to face off for a semi-final berth on Friday.
Ostapenko needed less than an hour to get past Paula Badosa, who caused an upset when the pair met at last year's delayed French Open.
There was to be no Roland Garros repeat for Badosa, though, as she was swept aside 6-2 6-2.
The sixth-seeded Ostapenko started her campaign impressively in Russia, producing 26 winners and just eight unforced errors. Her serve was also in good order, never giving her opponent a chance of a break.
Sasnovich had to work a little harder, rallying from a set down to see off Ana Bogdan 2-6 6-2 6-1. Her reward is a clash with third seed Fiona Ferro, who received a bye through to the second round.
Zvonareva, meanwhile, progressed in straight sets against Arina Rodionova, the Russian dropping just four games during a contest that lasted one hour and 22 minutes.
Victory was sealed at the second opportunity, a sliced backhand into the net by her opponent sending Zvonareva through to the last 32.
The American pulled out of the French Open due to back and neck injuries that have kept her out since April but made her return with a 6-2, 6-2 win in just 58 minutes.
Sasnovich matched Pegula in the opening four games, with both players trading points, but the world number five soon gained her foothold to cruise through the rest of the first set.
Pegula remained in control in the second, despite Sasnovich's late attempt at a comeback, but the Belarusian could not do enough to stop the top seed from advancing.
Pegula will now face Aleksandra Krunic or Jessika Ponchet in the next round.
Data Debrief: Pegula puts injury worries behind her
Pegula last played in America's Billie Jean King Cup victory in April, but she showed no signs of rustiness on her return to the court.
She showed no signs of rustiness though, hitting nine winners and just six unforced errors on her way to sealing her victory.
In a close contest, Belarus' Sasnovich took advantage in the big moments, winning five of her 10 break point opportunities, while Sorribes Tormo could only convert two-of-nine.
While seventh seed Sasnovich was the highest-seeded winner on the day, fifth seed Irina-Camelia Begu fell victim to a shock 6-3 6-2 upset from world number 415 Sofia Kenin.
Kenin, a wildcard, now advances to the quarter-finals after winning two matches in a row for the first time since January, and prior to this week she had lost her previous nine matches.
Alize Cornet will likely meet second seed Martina Trevisan in the quarter-final after a strong 6-3 7-6 (7-1) triumph against Denmark's rising 19-year-old talent Clara Tauson.
Liudmila Samsonova needed only 59 minutes to breeze past Iryna Shymanovich 6-1 6-0, and it was similarly smooth sailing for Magda Linette as she defeated Sorana Cirstea 6-4 6-2.
Meanwhile, at the Granby Championships in Canada, only one match was able to finish before the rain halted the rest of the day's play.
Spain's Nuria Parrizas-Diaz was too much for England's Harriet Dart, winning 6-2 6-4, while top seed Daria Kasatkina will just need to add the finishing touches when she resumes her match against Magdalena Frech, leading 6-3 5-0.
Sasnovich reached round three at Flushing Meadows in New York, thrashing 2019 French Open runner-up Marketa Vondrousova in her second match before bowing out to Yulia Putintseva.
Back to regular WTA Tour action, swapping US hardcourts for clay in Turkey, 106th-ranked Belarusian Sasnovich began slowly in her clash with Zarina Diyas before improving to beat the Kazakh seventh seed 3-6 7-5 6-1 on Tuesday.
Polona Hercog, the Slovenian third seed, saw off Turkish hope Berfu Cengiz 6-2 6-3, with wins also for Spain's Paula Badosa and Sara Sorribes Tormo, the latter going through after British fifth seed Heather Watson retired when trailing 3-2 in the opening set.
Italy's Jasmine Paolini and Slovakia's Anna Karolina Schmiedlova came through tight three-set contests, with Canada's former Wimbledon finalist Eugenie Bouchard among the players to battle through qualifying early in the day to reach the main draw.
Svitolina – the fourth seed – dropped the first set to 32nd seed Sorana Cirstea before fighting back to reach the round of 16 at the WTA Premier 1000 event on Sunday.
Two-time grand-slam champion Azarenka joined Svitolina in the next round after knocking out seventh seed Petra Kvitova, but 11th seed Halep became Aliaksandra Sasnovich's latest conquest.
SVITOLINA SURVIVES CIRSTEA SCARE
Svitolina needed two hours, 32 minutes to outlast Cirstea 4-6 6-4 7-6 (7-3) and remain unbeaten in three career matches against the Romanian.
The Ukrainian was on the defensive throughout the opening set, facing 11 break points on her serve and managing to save eight of them before Cirstea finally won out.
Svitolina tightened things up from there, saving four of five break points the rest of the match.
"It was a very tough match today, and I was fighting and trying to find my game," said Svitolina, who will face Jessica Pegula next. "It was a bit of a rollercoaster.
"I wish I could play a little bit better in the first set, I had chances to grab that set, but unfortunately it didn't play the way I wanted. I had to fight for every point, and Sorana played a great match, I think. I'm happy that I could win today."
AZARENKA TAKES DOWN KVITOVA
Azarenka won her first two Tour-level matches against Kvitova in 2008 and 2009 but had prevailed only once in six meetings since then before Sunday's 7-5 6-4 triumph.
The two-time Indian Wells champion converted break-point chances when she needed them, six of 11 in all, and had 18 unforced errors to Kvitova's 26.
"It was important to just stay there, really take my opportunities, not to let her," Azarenka said after taking down two-time Wimbledon winner Kvitova. "If she gets in the groove in couple points, not to kind of let her extend that streak, if you want to call it that.
"I was trying to still create opportunities for myself, be more aggressive, and honestly just believing also that what I'm doing is right and see how I can execute that. So intention was good. Execution followed after."
ANOTHER UPSET FOR SASNOVICH
After knocking off US Open champion Emma Raducanu in the second round, Sasnovich ushered out another grand slam winner in Halep 7-5 6-4 and will next face Azarenka.
The world number 100 had 22 winners to 16 for Halep and won 61.8 per cent of points on her serve as she reached the round of 16 at Indian Wells for the first time.
In other matches Sunday, ninth seed Anastasia Pavlyuchenkova fell to 23rd seed and US Open runner-up Leylah Fernandez 5-7 6-3 6-4, while second seed Iga Swiatek dropped only one game in a 6-1 6-0 demolition of Veronika Kudermetova.
Jelena Ostapenko and Shelby Rogers also advanced.
The world number one had won the first set 7-5, but her Australian opponent was forced to concede at 2-2 in the second.
Swiatek is now the female player with the most wins in a single year (58) in the last five seasons. Former world number one Ash Barty recorded 57 victories in 2019.
The second seed in Ostrava is out after Paula Badosa was beaten 7-6 (7-4) 6-4 by Petra Kvitova in front of a delighted Czech crowd, while Elena Rybakina is also through to the last eight after coming from behind to win against Aliaksandra Sasnovich 6-7 (5-7) 6-4 7-5.
Wednesday's two other matches saw more success for Czech participants as Karolina Muchova beat seventh seed Beatriz Haddad Maia 6-4 6-4, while Barbora Krejcikova defeated Shelby Rogers 6-2 6-2.
There was less Czech joy at the Jasmin Open in Tunisia, where Katerina Siniakova was beaten 7-5 6-2 by Claire Liu.
Elsewhere, number three seed Alize Cornet eased past Harriet Dart 6-3 6-4, Elise Mertens came from behind to beat Despina Papamichail and Diane Parry went through against French compatriot Lucrezia Stefanini after the latter retired hurt in the second set.
