Petrova countered 71 unforced errors with seven aces. She was coming off a "stress reaction'' injury to her right foot and played with both feet taped.
She will play on Saturday against fifth-seeded Aleksandra Wozniak of Canada, who outlasted Tamira Paszek of Austria 6-3, 3-6, 7-6 (7).
Petrova fell behind 3-0 in the third set before rallying to tie it 6-6.
The world No. 10 took a 6-3 lead in the tiebreaker, but Bondarenko hit a forehand winner and Petrova put a forehand into the net to make it 6-5. Bondarenko lost when she hit a backhand into the net.
Wozniak blew a 5-2 lead in the third set and double-faulted on her first match point before ousting Paszek.
The Canadian, nursing an injury to her right shoulder, led 6-5 in the tiebreaker before the double fault. After an exchange of points, Paszek put two balls wide to give Wozniak the win.
"I got tight and that's what led to the double fault,'' Wozniak said. "But, you have to fight your nerves and just stay strong, especially in a tiebreaker.''
Petrova holds off Bondarenko at Ponte Vedra
Petrova holds off Bondarenko at Ponte Vedra
