Medvedev failed to convert three match points in the second set after taking a 5-2 lead. An hour later the Russian was still playing, and on a humid afternoon with temperatures in the mid 80s, he paid a price.
Second-ranked Osaka, the reigning US and Australian Open champion, will next face the winner of a later match between 16th seed Elise Mertens and 22nd-seeded Anett Kontaveit.
Despite conceding the first set, Russia's Kuznetsova put up a spirited defence in the second leading to increasingly sprawling rallies with her well-matched Belarusian opponent.
Kasatkina was the first winner of the new season, beating Wang Qiang 6-2, 3-6, 6-2 after trailing 2-0 in the deciding set. It was the Russian's first match since October, and Wang's first since March.
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Naomi Osaka, who won the US Open two years ago, advanced to the quarterfinals by beating No. 14 seed Anett Kontaveit 6-3, 6-4.
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It was Ferro’s second WTA title and the 23-year-old sealed it on her second championship point when Kontaveit hit a return long.
Palermo Ladies Open: Petra Martic to face Anett Kontaveit, Fiona Ferro against local favourite Camila Giorgi in semi-finals
Martic beat qualifier Aliaksandra Sasnovich 7-6 (5), 7-6 (3) then Kontaveit defeated Italian teenager Elisabetta Cocciaretto 6-1, 4-6, 6-1.
Petra Martić got past qualifier Liudmila Samsonova and will next face another qualifier, Aliaksandra Sasnovich.
Australian Open 2020: Garbine Muguruza sets up semi-final clash with Simona Halep after beating Anastasia Pavlyuchenkova
Fourth seed Simona Halep took just 53 minutes to thrash Estonian Anett Kontaveit 6-1 6-1 and storm into the semi-finals of the Australian Open on Wednesday.
Australian Open 2020: Garbine Muguruza, Simona Halep power into last-eight, Anett Kontaveit made to work
Garbine Muguruza’s participation at the Australian Open had been in doubt due to illness but Monday’s demolition of Kiki Bertens to reach the quarter-finals proved she is not only nearing full health but could also be a genuine title contender.
Australian Open 2020: Poland's Iga Swiatek shows off her full potential as she goes down swinging to Anett Kontaveit
Kontaveit struck 29 winners to 42 from Swiatek, but also made far fewer errors than her opponent – which proved to be the difference in the end.
Australian Open 2020: Rafael Nadal overcomes Nick Kyrgios challenge in four sets; Alexander Zverev enters quarter-final
Rafael Nadal has beaten Aussie hope Nick Kyrgios in four sets to reach the quarterfinals at a Grand Slam tournament for the 41st time.
Australian Open 2020: Anett Kontaveit’s third round demolition show of Belinda Bencic a case of 'see it to believe it'
Kontaveit won the first nine games of the match against Bencic, and dropped just one game overall. She won 86% of her first serve points, 63% on her second serve, and never faced a break point. She hit 21 winners and just seven unforced errors
Australian Open 2020: Karolina Pliskova, Belinda Bencic join exit queue, Simona Halep, Angelique Kerber avoid chaos
Second-seeded Karolina Pliskova and No. 6 Belinda Bencic went the same way as Serena Williams and defending champion Naomi Osaka by exiting the Australian Open in a chaotic third round.
Australian Open 2020: Serena Williams secures berth in third round; Coco Gauff to meet Naomi Osaka after three-set win
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Cincinnati Masters: Novak Djokovic cruises into quarters but Roger Federer crashes out; Ash Barty, Naomi Osaka win
Novak Djokovic has a clear road to the Cincinnati Masters final after producing an efficient victory while Roger Federer crashed out of the US Open tune-up event.
Rogers Cup: Serena Williams, Naomi Osaka set up rematch of 2018 US Open final; Karolina Pliskova cruises into quarter-finals
Serena Williams and Naomi Osaka booked a rematch of last year's US Open final in the Rogers Cup quarter-finals after straight-set triumphs.
Rogers Cup: Maria Sharapova suffers defeat in first round against Anett Kontaveit; Daria Kasatkina downs Angelique Kerber
Anett Kontaveit spoiled Maria Sharapova's return from injury on Monday, rallying to beat the five-time Grand Slam champion 4-6, 6-3, 6-4 at the WTA tournament in Toronto.