Stosur saved two match points in the tense conclusion before converting a second match point of her own for a gritty 6-4 2-6 7-6 (9-7) second-round win.
The Australian progresses to the quarter-finals, where she will meet eighth seed Bojana Jovanovski.
Stosur, the world No.17, came into the match with only a scrappy three-set win over qualifier Madison Brengle from four matches this season.
The 2011 US Open champion said she'd improved with each match since losing all three at the Hopman Cup, and she showed that against the massive forehand firepower of the tall Mladenovic.
Stosur was broken in the first game of the match but then broke Mladenovic twice to lead the first set 5-3.
The 20-year-old Frenchwoman broke again for 5-4 but her serving let her down and a double fault gifted the set to the Australian.
Mladenovic got the early break again in the second and led 5-2 before clinching the set with an ace.
Stosur broke early to lead the third 2-0 but was immediately broken by Mladenovic and in a tense set both players held until the nailbiting decider.
Earlier, defending champion Elena Vesnina retired form her second-round match against Spanish qualifier Estrella Cabeza Candela with a hip injury.
The Russian world No.24, who made the fourth round at the Australian Open last year, quit with the score 4-6 6-2 4-1.
World No.114 Cabeza Candela will play 2012 Hobart champ Mona Barthel or Romanian Monica Niculescu in the quarter-finals.
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