The Belgian, who was four match-points down in the second set, was able to come from behind to beat the fifth seed and progress to the final eight, despite injuring her ankle in the opening set.
"I can't believe I won," she said. "It knew before the match that it was going to be tough, that it was going to be physically and mentally tough, but obviously I didn't expect this.
"It was amazing to get through, to fight with probably not my best tennis. You just want to try to win each point and not give up."
Clijsters will face either Caroline Wozniacki or Jelena Jankovic in the next round.