Playing in the first match on Court Philippe Chatrier, the eighth-seeded Stosur dropped her serve in the opening game but then reeled off four straight games to take the lead for good.
The Australian reached the final at Roland Garros last year but lost to Francesca Schiavone. In 2009, Stosur reached the French Open semifinals.
"Obviously I thought of last year's last match," Stosur said. "It was nice to get out there again and, of course, if you can play on any court here you'd want to play on that one."
"I don't think I have to necessarily prove anything. I think having made semis and a final, I think that kind of proves that I have been able to back it up and it wasn't a fluke the first time," Stosur said. "Of course I'd love to go a little bit further this year, but I think it's more just wanting to achieve that rather than I have to do it because of X, Y, and Z."
"It's never easy, especially in the first round," said Jankovic, who beat Alona Bondarenko of Ukraine 6-3, 6-1. "Today was a little bit windy, so the ball was swirling around. But I just played the best I could in these kind of circumstances and did my job."
The French Open is the only one of the four Grand Slam tournaments to start on a Sunday, and there were 32 matches scheduled on eight courts.