Night matches were postponed due to the heavy rain in Paris. Play was halted by rain for nearly three hours at one point during the day.
Kvitova, a two-time Wimbledon champion from the Czech Republic, appeared en route to an easy victory on the women's side after winning the first set and holding a 3-0 lead in the second. But Kovinic, from Montenegro, battled back to win six of the next seven games to tie the match and put up a good fight in the final set before Kvitova prevailed.
"I think that in the second set I missed four easy backhands and I felt like that was kind of the turning point and I was kind of down mentally after this game," Kvitova told reporters after the match. "I think she just came back and tried to play more aggressive.
"I was just, I think, running like too far from the baseline. I was just trying to put the ball in, but it's not really my game."
The Czech Republic's Lucie Safarova, seeded 11th, had a much easier time by cruising to a 6-0, 6-2 victory over Russia's Vitalia Diatchenko.
The match involving Safarova, the runner-up in last year's French Open, was one of those that was stopped by the lengthy rain delay and later completed.
In other women's matches, Russia's Anastasia Pavlyuchenkova defeated Spain's Sara Sorribes Tormo 6-2, 6-0; Switzerland's Viktorija Golubic beat American Alison Riske 6-2, 1-6, 6-2, and Taipei's Su-Wei Hsieh was a 7-6 (6), 6-3 winner over Spain's Lara Arruabarrena.
One the men's side, Australian Nick Kyrgios received a code warning for shouting at a young ball retriever and was tested by Italy's Marco Cecchinatio before finishing off a 7-6 (6), 7-6 (6), 6-4 victory.
The 17th-seeded Kyrgios was admonished by chair umpire Carlos Ramos for the tone of voice he used.
"I think with someone like me that's pretty emotional it can frustrate me a little bit, but I felt like I dealt with it pretty well," Kyrgios said afterward. "Focus in that first set is obviously very crucial to win in a best-of-five match.
"For me that time I didn't get mad at all. I just said it a little bit loud. The crowd was going on. I just felt like it was a bit rough."
Fifth-seeded Kei Nishikori of Japan was one of the players who had a match halted by the rain. Nishikori holds a 6-1, 7-5, 2-1 lead over Italy's Simone Bolelli when play resumes Monday.
In other completed men's matches, 19th-seeded Benoit Paire of France was a 6-2, 4-6, 6-4, 1-6, 6-4 winner over Radu Albot of Moldova; Igor Sijsling of the Netherlands defeated Romania's Adrian Ungur 6-1, 6-2, 7-6 (5); Russia's Teymuraz Gabashvili recorded a 7-6 (1), 6-2, 6-3 victory over American Donald Young, and Russia's Andrey Kuznetsov beat Germany's Benjamin Becker 6-2, 6-2, 6-4.