Sloane Stephens into 3rd round
No. 6 Sixth-seeded Jelena Jankovic won 7-5, 6-0 against Japan's Kurumi Nara, and No. 10 Sara Errani of Italy -- a runner-up in 2012 and semifinalist last year -- won 6-2, 6-4 against Germany's Dinah Pfizenmaier.
The 11th-seeded Ivanovic dropped her own serve three times but broke Svitolina's serve six times. She clinched victory on her second match point with a forehand winner down the line on Svitolina's serve.
Since winning here in 2008 for her only Grand Slam title, the 26-year-old Ivanovic has reached only one quarterfinal at a major -- the US Open in 2012 -- and has never been past the fourth round at Roland Garros.
No. 23 Lucie Safarova of the Czech Republic also advanced. She will next play Ivanovic.
Bertens next plays Spaniard Silvia Soler-Espinosa, who downed Belgium's Yanina Wickmayer 6-2, 6-4.
The Associated Press contributed to this report.