Sam Stosur advances in Moscow
Stosur, coming off her second title this season and fifth of her career, converted four of 11 break opportunities.
"Kaia was a really dangerous opponent," Stosur said. "I really tried to start the match well, on top."
A finalist last year, Stosur will next play Alize Cornet.
The 23rd-ranked Kuznetsova overcame an early break in the second set and earned the victory on her third match point.
"I was hurrying a bit and wanted to play more aggressive, thus making more mistakes," Kuznetsova said. "But I can say that now, by the end of the season, I play much better."
Kuznetsova missed Wimbledon, among several tournaments in midseason, because of an abdominal injury. She will try to reach the semifinals for only the second time in Moscow when she faces second-seeded Roberta Vinci of Italy, who beat Elena Vesnina 2-6, 6-1, 6-4 after a first-round bye.
Vesnina, who won her first two singles titles this season, broke twice in the opening set. But Vinci broke the Russian three times in the second to force a decider. The 11th-ranked Italian broke decisively in the seventh game and served out.
In the men's second round, defending champion Andreas Seppi broke Paolo Lorenzi twice in the second set to win 6-3, 6-2. In the quarterfinals, the second-seeded Seppi will face Edouard Roger-Vasselin, who ousted Sergiy Stakhovsky 7-5, 7-5.
The 2011 champion Janko Tipsarevic was upset by Russian wild card Karen Khachanov 6-4, 6-4. The 17-year-old Khachanov advanced to his first career ATP quarterfinals in hitting 14 aces past third-seeded Tipsarevic.
Playing his fourth tour match, Khachanov broke Tipsarevic once in each set and didn't offer a break chance.
"It was a tight match. I was trying, but Karen didn't give many chances on his serve," Tipsarevic said. "Two chances he had on my serve, he broke me. He plays really well for his age, and if he improves his fitness he has all the potential to become a very good player."
Earlier, Alisa Kleybanova advanced to her first quarterfinal since being treated for cancer, beating the sixth-seeded Carla Suarez Navarro 7-6 (2), 6-4.
Kleybanova was diagnosed with blood cancer and underwent chemotherapy until December 2011. The 24-year-old Russian was ranked as high as No. 20, but is 254th after missing much of the last two seasons.
She resumed her career in May and won a challenger event before returning in August to the WTA Tour in Toronto.
"I always liked to play with Carla, and I knew it would be a tough match today," Kleybanova said. "I played more accurate on key moments and that's why I won."
Anastasia Pavlyuchenkova, who is looking for her third title of the season, beat Russian compatriot Maria Pavlyuchenkova 6-3, 6-3 to reach her seventh quarterfinal.