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.
Samantha Stosur opened her account by beating Kaia Kanepi of Estonia 6-3, 6-1. Stosur, coming off her second title this season and fifth of her career, converted four of 11 break opportunities.
"I played really well and Kaia was a really dangerous opponent," Stosur said. "I really tried to start the match well, on top."
Last year's runner-up, Stosur has Alize Cornet of France next.
Meanwhile, 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 quarterfinals since being treated for cancer, beating the sixth-seeded Carla Suarez Navarro of Spain 7-6 (2), 6-4.
The Russian was ranked as high as 20th, but after missing much of the last two seasons because of her illness is now 254th. She resumed her career in May and won a challenger before returning to the tour in Toronto in August.
"I always liked to play with Carla and I knew it would be a tough match," Kleybanova said. "But I think that today I played more accurate on key moments and that's why I won."
Kleybanova was diagnosed with blood cancer and underwent chemotherapy until December 2011.
Anastasia Pavlyuchenkova, who is looking for her third title this year, beat Russian compatriot Maria Kirilenko 6-3, 6-3.
