Vesnina defeated Kuznetsova 6-4, 3-6, 6-0 in 1 hour, 41 minutes in the second-round match. Ayumi Morita of Japan defeated the 15th-seeded Anna Chakvetadze 7-5, 6-2 and ninth-seeded Agnieszka Radwanska of Poland lost to her younger sister, Urszula, 6-4, 6-3 in first-round matches.
Amelie Mauresmo, winner of last week's Paris Open, withdrew at the last minute and was fined $7,500 by the WTA.
Vesnina, ranked 75th in the world and a qualifier here, was thrilled with her victory.
"She's obviously a top 10 player, so it's even more special," the 22-year-old Russian said. "I think I really troubled her today. She wasn't comfortable at all and I feel she was totally lost in the third set. She kept looking at her coach."
Kuznetsova said the fast court was a problem.
"The court was fast, and I just could not get the feel of the ball throughout the match," Kuznetsova said. "I am now off to Moscow, and I really need to work hard on my game."
The event has attracted a quality field this year with most of the top 10 competing. Australian Open winner Serena Williams is the top seed.