Cornet, the tournament's No. 3 seed, got past Karin Knapp 1-6, 7-5, 6-0 in the day's final match. Knapp controlled the first set but Cornet used three breaks to take the second set before running away with the third.
She won more than 60 percent of the points both serving and receiving in the final-set shutout.
Chanelle Scheepers also moved into the third round by taking out Maria-Teresa Torro-Flor 6-2, 6-1. She finished off seven of her eight break opportunities on the day.
No. 6-seeded Francesca Schiavone was a 7-6 (7-4), 6-4 winner over Petra Martic in a first-round match.
Other first-round results Wednesday saw Olga Puchkova get past Daniela Hantuchova 3-6, 7-6 (7-5), 7-5; Flavia Pennetta advance over Melinda Czink 6-1, 7-6 (7-2); and Yulia Putintseva defeated Mathilde Johansson 6-3, 6-4.