No. 2 Cornet overcame nine double faults and seven service breaks. The only seed to reach the second round, she next meets Taiwanese wild card Hsieh Su-Wei, the 2012 champ.
Hsieh beat Chinese wild card Zhu Lin 6-4, 6-2.
Another Chinese wild card, Wang Yafan, who stunned top seed Samantha Stosur in her first WTA main draw match on Tuesday, defeated Croatian qualifier Petra Martic 7-6 (5), 6-3. Wang's quarterfinal opponent will be countrywoman Zhang Kai-Lan.
In other quarterfinals, Romanian Monica Niculescu will play Puerto Rico's Monica Puig, and Switzerland's Timea Bacsinszky faces Spain's Maria-Teresa Torro-Flor. All four have one career WTA title, with Puig and Torro-Flor winning their first this year.