Sharapova, the world No. 2 and tournament defending champion, had 34 unforced errors -- 22 in the second set. That was more than half of the points Zheng won (58).
But Sharapova was 8-for-8 in break opportunities, posting four breaks in each set and winning 61 percent of the points off Zheng's serve.
The third-seeded Azarenka was on the court nearly 2 1/2 hours in getting by Cornet. She had to overcome 10 double faults and six losses of serve but she controlled the third set, taking more than 60 percent of the points both serving and returning.
Sharapova next goes against 17th-seeded Sloane Stephens, who was a 6-4, 6-7 (5-7), 6-3 winner over Marina Erakovic, while Azarenka draws Francesca Schiavone, who ousted No. 13-seeded Marion Bartoli 6-2, 6-1. Schiavone won the French Open in 2010 and was the runner-up in 2011.
Azarenka will face 2010 French Open winner Francesca Schiavone, who staged 6-2, 6-1 upset of 13th-seeded Frenchwoman Marion Bartoli. The Italian improved to 18-2 in Paris in her last four appearances.
Bethanie Mattek-Sands advanced to the fourth round of a major tournament for the second time in her career after she beat Paula Ormaechea 4-6, 6-1, 6-3.
Other third-round winners Saturday included American Jamie Hampton and 12th-seeded Russian Maria Kirilenko.
Notable deaths of 2014 [PHOTOS]