Kuznetsova beat French qualifier Mathilde Johansson in a tense match, 6-7 (5-7), 7-5, 6-3, while Chakvetadze defeated Stephanie Dubois (Canada) 2-6, 6-1, 8-6.
Marat Safin had an easy win over the Italian, Fabio Fognini, 6-1, 6-2, 7-6 while Igor Andreyev beat Jiri Vanek 6:7 (6:8), 7:6 (7:2), 6:2, 6:2.
The unseeded Russian Anastasia Pavlyuchenkova beat Alize Cornet (France) 7-6, 7-6.
Evgeniya Rodina overcame the Frenchwoman Virginie Razzano 0-6, 6-2, 6-4. Elena Vesnina beat Spain's Nuria Llagostera Vives 6-7, 6-4, 6-4. In a battle between two Russians, Vera Dushevina knocked out Maria Kirilenko, seeded 19, 6-3, 6-3.
Maria Sharapova, the No. 3 seed, opens her Wimbledon today with a match against the unseeded French player Stephanie Foretz. Sharapova, one of the favorites for the women's singles, has said she will play her first game in shorts.
British police have banned 11 individuals believed to be 'fixated' on certain tennis players from the tournament, which runs until July 6.