The fourth-seeded Kuznetsova, who is the highest seed still playing in the event, grounded Japan's Ai Sugiyama, 6-2, 7-6 (8-6).
Kuznetsova, who lost to Justine Henin-Hardenne in the French Open finale in June, has reached at least the quarterfinals in 12 of her 14 tournaments this year.
No. 9 seed Dinara Safina of Russia beat American Shenay Perry, 7-5, 6-3; Anna Chakvetadze of Russia topped France's Nathalie Dechy, 6-1, 6-4; and 14th-seeded Katarina Srebotnik of Slovenia beat Stephanie Dubois of Canada, 6-3, 6-2.
Fifth-seeded Nicole Vaidisova of the Czech Republic was one of two players to withdraw on Thursday.
Celebrity Families of 2014 [PHOTOS]