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 two quarterfinals at a major and has never been past the fourth round at Roland Garros.
She next faces No. 23 Lucie Safarova of the Czech Republic, who won 6-1, 5-7, 6-3 against Casey Dellacqua of Australia.