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.
Two-time Grand Slam champion Svetlana Kuznetsova advanced to the quarterfinals by beating Sofia Arvidsson 6-2, 7-5.
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."
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.
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.