Stosur, the 2011 U.S. Open champion, rallied after having her serve broken three times in the first set at the Utsubo Tennis Center. Razzano had 10 double-faults.
The Australian will next face Jamie Hampton in the quarterfinals. The American defeated wild-card entry Tamarine Tanasugarn of Thailand 7-5, 7-5.
Also, Japanese qualifier Misaki Doi beat Olga Puchkova of Russia 6-3, 6-2 and will next take on Chanelle Scheepers of South Africa, while Kai-Chen Chang of Taiwan defeated Casey Dellacqua of Australia 7-6 (4), 6-3 and will next face eighth-seeded Laura Robson of Britain in the quarterfinals.