After the singles rubbers were split on day one of the final, Kristina Mladenovic gave France the advantage with a 2-6, 6-4, 7-6(1) victory over Ashleigh Barty.
The French then opted to field Pauline Parmentier over Caroline Garcia for the second singles match of the day, but she lost out to Ajla Tomljanovic in straight sets.
It all came down to the decisive doubles rubber, in which Garcia and Mladenovic rekindled the partnership that brought them the Ronald Garros title in 2016, beating Barty and Stosur 6-4, 6-3.
"I’m not sure I’ve completely realized, because it’s been a crazy weekend for everyone," Garcia said after the win.
"The doubles went very fast, but we played every point so focused, and we played a great match and it was very tense."