Kvitova, playing her opening match after a first-round bye, went from trailing 2-1 in the first set to a 3-0 lead in the second. But she double-faulted on match point at 5-1 to let Azarenka back into the match.
In the quarterfinals, Kvitova will face eighth-seeded Donna Vekic, who beat lucky loser Veronika Kudermetova 6-4, 6-3. Maria Sharapova withdrew from the tournament with a right shoulder injury, hours before she was to play Daria Kasatkina in the second round. Sharapova had beaten Australia's Daria Gavrilova in the first round Monday and was aiming to win a WTA event in her native Russia for the first time.
"I tried to do everything to be ready to play this week but my right shoulder is still bothering me," she said on the tournament website. "In the next few days I'll meet my medical team to evaluate the situation and I'm looking forward to getting back into competition as soon as possible. Playing in front of my home crowd was something special for me and it's a shame that I couldn't keep competing at this fantastic tournament."
A walkover for Kasatkina sets up a quarterfinal with Russian veteran Vera Zvonareva, who upset fifth-seeded Julia Goerges 4-6, 6-4, 6-4 despite being ranked 97th, 81 places behind her German opponent.
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