But defending champion Maria Sharapova's scheduled last-16 clash against Lucie Safarova was pushed back until Monday after rain caused a lengthy stoppage at a chilly Roland Garros.
The Russian second seed now faces the prospect of playing matches on two successive days with the first two women's quarter-finals already programmed for Tuesday.
Seventh-seeded Serb Ivanovic defeated Russian ninth seed Ekaterina Makarova, a semi-finalist at the Australian Open in January, 7-5, 3-6, 6-1 to book her eighth career Grand Slam quarter-final.
The 20-year-old Svitolina beat fellow former Roland Garros junior champion Alize Cornet on a windy, chilly Court Philippe Chatrier 6-2, 7-6 (11/9).
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"It's amazing to be in the quarter-finals again. To be honest, coming into the tournament I didn't really expect that at all," said Ivanovic.
"But I really worked hard for each match. I worked hard even before the tournament to reach the quarter-final again."
It was Svitolina's first win over Cornet in three meetings and the result ended French hopes in the women's singles for another year.
She displayed nerves of steel to achieve victory, failing to serve out the tie in the 10th game of the second set and then allowing five match points to slip through her fingers.
However, she secured victory on her sixth match point when Cornet went long with a backhand, her 42nd unforced error of the windswept tie.
"I hope I can serve better, of course, because I was serving not so good against her in Madrid," said Svitolina.
With four men's last-16 ties still to be played on Court Philippe Chatrier and Court Suzanne Lenglen, it was decided that Sharapova and Safarova will play their fourth round on Monday instead.
Sunday's other fourth round clash between 33-year-old Italian Flavia Pennetta, who has never played in the French Open quarter-finals, and Garbine Muguruza, the Spanish 21st seed, was also shelved.
"The scheduled 4R matches between Sharapova-Safarova & Muguruza-Pennetta have been cancelled for today," tweeted the organisers.