Burdette led 5-2 and was serving for set point when the 10th-ranked Bartoli fired off a series of winners.
In other second-round matches, Urszula Radwanska of Poland upset eighth-seeded Marina Erakovic of New Zealand 6-3, 3-6, 6-4, and Sorana Cirstea of Romania beat lucky loser Zheng Saisai of China 6-3, 6-3.