The 17th-seeded Stephens served for the match at 5-3 in the third set, but got broken.
Stephens broke serve at love in the final game to win the match, which lasted 2 hours, 36 minutes.
Stephens is one of nine women to reach the second week at the 2013 Australian and French Opens and will make it that far at Wimbledon with one more victory.
She played her first Grand Slam semifinal at the Australian Open in January after upsetting Serena Williams. At the French Open, she made it to the fourth round.