Mauresmo led 4-1 in the tiebreaker at Ariake Colloseum, but Cibulkova fought back and took a 9-8 lead with a crosscourt forehand before winning when Mauresmo's return went long.
"The match was really close in the third set," Cibulkova said. "I should have closed it out earlier ... but I fought through and was able to win the match."
Mauresmo blew four match points, including one in the tiebreaker.
"It's always frustrating to lose a match when you have match points and can't close it out," Mauresmo said. "I let her get back into the match in the second set and it's disappointing to lose like that."
In other first-round matches, Francesca Schiavone of Italy defeated seventh-seeded Daniela Hantuchova of Slovakia 6-1, 7-5, and Kaia Kanepi of Estonia defeated eighth-seeded Anna Chakvetadze of Russia 6-4, 3-6, 6-3.
Also, Marion Bartoli of France cruised to a 6-2, 6-2 win over Maria Kirilenko of Russia, Flavia Pennetta of Italy downed Israel's Shahar Peer 6-4, 6-4, and Japan's Ayumi Morita defeated Agnes Szavay of Hungary 6-7 (7), 7-5, 6-4.