Kvitova landed just 51 percent of her first serves, but was strong on the second serve as Vesnina failed to make an impact.
Garcia, ranked seventh in the world, lost to Russian qualifier Elena Rybakina 4-6, 7-6 (6), 7-6 (5).
Last year’s French Open champion Jelena Ostapenko outplayed Russian veteran Vera Zvonareva to reach the quarterfinals, winning 7-6 (6), 6-3. She’ll play the winner of the match between Kvitova and Begu.
In an all-Russian first-round match, sixth-seeded Anastasia Pavlyuchenkova lost 6-4, 6-4 to compatriot Ekaterina Makarova. Eighth-seeded Darya Kasatkina reached the quarterfinals with ease, beating Alize Cornet 6-2, 6-1.