The 18th-seeded Jankovic, a three-time semifinalist at Roland Garros, sealed victory on her fourth match point with an ace, shortly after Stosur missed a chance to break back to 5-5.
Three years ago, Stosur beat Jankovic in the semis to reach the Roland Garros final. This time, the 2011 U.S. Open champion took control of the opening set by breaking Jankovic's serve on her first try.
But Stosur dropped her serve twice in the second set and three times in the third.
Jankovic leads Stosur 7-2 overall and 2-1 on clay. She next plays American Jamie Hampton, who earlier upset seventh-seeded Petra Kvitova 6-1, 7-6 (7).