Jankovic fell behind early but rallied past Muguruza 6-3, 6-0 shortly before showers interrupted the first matches of the day on 12 other courts.
A brief dry spell allowed 2010 runner-up Sam Stosur to complete a second-round victory over Kristina Mladenovic, 6-4, 6-3, before the rain resumed.
Zheng Jie beat Melanie Oudin of the United States 6-3, 6-1. No. 27 Yaroslava Shvedova lost to Paula Ormaechea 6-4, 7-6 (6).
With the midday temperature in the mid-50s, Jankovic took the court wearing pink leggings and a long-sleeve pink shirt. She looked uncomfortable at the start, losing the first three games, then won the rest after adjusting to the strokes of Muguruza, a 19-year-old Spaniard.
"She's a great up and coming player," Jankovic said. "She hits the ball very hard, very flat. And especially in the beginning of the match she gave me a lot of trouble. I figured out the way to play against her, and I think I did pretty well."
Jankovic won with steady play from the baseline, while the big-swinging Muguruza committed 29 unforced errors.
Jankovic, seeded 18th, is a three-time semifinalist at Roland Garros. She'll next play 2011 U.S. Open champion Stosur.
The No. 9-seeded Stosur never lost serve but needed nearly an hour to take the first set against Mladenovic. Stosur reached set point for the fourth time on a fluky point with her opponent at the net, hitting a line-drive reflex volley that sent Mladenovic ducking to avoid. The ball kissed the baseline for a winner, and Stosur waved in apology to Mladenovic.
"I ran in thinking she would drop-shot, and she didn't drop-shot," Stosur said. "I don't know what grip I had, what I did, where it landed. I don't know anything. It was very, very lucky, but obviously at a really important time."
Said Mladenovic: "Obviously it was not on purpose, but it was an amazing reflex. I think she won't do it again in her life. We both laughed after it, because it was just an unbelievable one."
Mladenovic then double-faulted to lose the 16-point game. Stosur took a 3-0 lead in the second set and held the rest of the way.
Oudin, a former U.S. Open quarterfinalist, won only 19 of 51 points on her serve and held just once. Her elimination left six U.S. women still in draw midday through the second round.
No. 16 Philipp Kohlschreiber moved into the men's third round when Yen-hsun Lu withdrew because of a right ankle injury. Victor Hanescu also advanced when Dmitry Tursunov retired with a strained left thigh muscle trailing 6-4, 6-6 (3).
Play resumed in the afternoon after the second delay.
On Wednesday, No. 1-ranked Serena Williams extended her career-high winning streak to 26 matches — and improved to 69-3 since a first-round loss at the French Open a year ago — by beating French wild-card entry Caroline Garcia 6-1, 6-2.
"She's very accurate in her shots," Garcia said, "despite the fact that they're very hard."
When Garcia won the second point of the match on a missed forehand by Williams, someone in the crowd yelled, "Allez, Caroline!" Not much to cheer for after that, though: Williams won the next three points with a 111 mph ace, a 100 mph ace, and a 101 mph service winner. In all, Williams hit six aces and won 32 of 39 service points to reach the third round.
"I came out strong," Williams said, "because I knew I had to."
The 19-year-old Garcia is ranked only 114th, but much is expected of her. Against Maria Sharapova in the 2011 French Open, she won the first set and led 4-1 in the second before collapsing completely, losing the next 11 games and the match. Her performance was good enough to inspire Andy Murray to write on Twitter that Garcia "is going to be No. 1 in the world one day."
For now, it's Williams who holds that distinction in the rankings, and she certainly looks like someone intent on keeping it that way.
"It's important for me to win easily," said the 15-time major champion, who won the French Open in 2002. "It's also important for me to play well. If I play well, it will bode well for me at Roland Garros."
Among the past Grand Slam title winners hoping to join her in the third round with victories Thursday: defending French Open champion Sharapova, Victoria Azarenka, Li Na and Petra Kvitova. There wasn't much time to rest for Azarenka and Kvitova, both of whom finally got to play on Wednesday and won first-round matches that had been postponed by rain.
In men's play, seven-time champion Rafael Nadal and top-ranked Novak Djokovic were scheduled to play second-round matches.