Cornet , seeded second, defeated Croatian Jelena Kostanic Tosic 7-5, 6-3 at the Fairmont Acapulco Princess. Zakopalova, the third seed, rallied past Spanish qualifier Maria Jose Martinez Sanchez 4-6, 6-3, 6-1.
Fourth seed Martina Muller of Germany was upset by France's Olivia Sanchez 6-2, 6-4.
No. 7 Estonian Kaia Kanepi ousted Spanish qualifier Arantxa Parra Santonja 6-4, 6-3 and No. 8 Romanian Edina Gallovits easily bested Mexican wild card Daniela Munoz Gallegos 6-1, 6-0. Fifth-seeded Italian Sara Errani defeated German Tatjana Malek 6-3, 6-3.
Other first-round winners were Czech Iveta Benesova, Romanian Sorana Cirstea, Czech Petra Cetkovska and Ukraine's Mariya Koryttseva.
The 2008 Acapulco champion will earn $28,700.