Jankovic, beaten by Stosur in the semi-finals 12 months ago, overcame Ukrainian Alona Bondarenko 6-3, 6-1 and will now face Russia's Vera Dushevina.
Jankovic, who also reached the Roland Garros last four in 2007 and 2008, rarely looked unsettled as she recorded a 13th victory over Bondarenko in 14 encounters.
The Serb spurned two break points but then broke for a 4-2 lead before taking the first set when Bondarenko scuttled a forehand into the net at 5-3 down.
A pair of breaks put the 26-year-old Jankovic 4-1 to the good in the second set and she overcame belated resistance from the Ukrainian to prevail on her third match point.
"I love playing at Roland Garros," Jankovic said.
"I've reached the semis three times. I really love the courts here and hopefully I can do well again.
"Obviously my dream is to win it and I will always do my best."
Kuznetsova, the 2009 champion and 13th seed this year, eased past Slovakian Magdalena Rybarikova 6-2, 6-3, while German dark horse Julia Goerges, seeded 19th, outclassed Swede Mathilde Johansson 6-1, 6-4.
Israel's Shahar Peer, the 19th seed, was the first seed to fall, going down 7-6 (7-4), 6-1 to Spain's Maria Jose Martinez Sanchez, who won last year's clay-court Italian Open.
Russia's 14th-seeded Anastasia Pavlyuchenkova beat Yaroslava Shevdova of Kazakhstan 7-5, 6-3, while Bulgarian 32nd seed Tsvetana Pironkova, a beaten semi-finalist at Wimbledon last year, defeated Australian wildcard Casey Dellacqua 7-5, 6-3 to reach round two.
Alize Cornet become the first French woman into the second round, the world number 80 seeing off Renata Voracova of the Czech Republic 6-4, 6-2.
Schiavone was to begin her title defence against American 19-year-old Melanie Oudin on Monday, with Danish top seed Caroline Wozniacki kicking off her campaign against 40-year-old Japanese veteran Kimiko Date Krumm.
-AFPTags: sport, tennis, france