Japan's Osaka, ranked 406th, beat world number 19 Stosur 4-6 7-6 (9-7) 7-5 in her maiden WTA Tour appearance, with the 16-year-old producing a stunning revival to down the 2011 US Open champion.
Stosur held match point in the second set tie-break, having fought back from a break down earlier in the set, only for Osaka to reel off three straight points to level the match.
The unseeded Australian regained the upper hand when she broke in the fifth game of the third set, and maintained her lead until she faltered when serving for the match at 5-4.
Osaka broke back to 30, and then broke Stosur again in the 12th game to cause one of the upsets of the 2014 season.
Next up for Osaka is eighth seed Andrea Petkovic, after the German dispatched of Croatia's Ajla Tomljanovic 6-1 6-4.
Rising Czech star Karolina Pliskova thrashed Japanese veteran Kimiko Date-Krumm 6-1 6-3, to earn a second-round meeting with returning world number one Serena Williams.
Americans Coco Vandeweghe, Sachia Vickery and Varvara Lepchenko all progressed past the first round.
Vandeweghe ousted fellow American and wildcard Kristie Ahn 6-2 7-5, while qualifier Vickery thrashed China's Zhang Shuai 6-1 6-2 and Lepchenko was too good for France's Caroline Garcia 6-4 6-3.