The fourth-seed Indo-Russian combination dismissed the challenge of American-Belarusian pair 6-2 7-6 (7) in the third round contest.
It was a good day for Vesnina as she earlier partnering India's Sania Mirza reached the women's doubles semifinals.
Bhupathi-Vesnina may run into the Sania-Rohan Bopanna Indian pair in the mixed doubles semifinals if both the pairs win their respective quarter-finals.
Bopanna and Sania are one step away from making the quarter-finals as they face Czech team of Martin Damm and Renata Voracova in the third round.
Earlier today, they scored a 7-5 6-2 victory over Jamie Murray of Great Britain and Jarmila Gajdosova of Australia in the second round.