Sharapova, the No. 1 seed in this event, came back for a 6-7 (9-11), 6-3, 6-2 victory. She had to overcome eight double faults in a match that lasted two hours, 43 minutes.
Sharapova, winner of this year's Australian Open, will meet Lindsay Davenport in the semifinals in a rematch of the Australian Open title match, which Sharapova won 6-1, 6-3.
Davenport ousted Agnes Szavay 6-4, 7-6 (7-3).
Cornet was a 6-4, 6-2 winner over Virginie Razzano and Cibulkova surprised Amelie Mauresmo 6-1, 7-6.