Mauresmo, who missed the Italian and German Opens because of a rib injury, looked shaky much of her time on center court.
"I had ups and downs, good things and some big mistakes, and that's also linked to very short preparation time," said Mauresmo, who won the Australian Open and Wimbledon in 2006. "But that's the kind of match that's nice to get through, and especially with such a third set.
"So now all I have to do is hope it gets better for my next match," she said.
When play resumed, No. 14 Agnieszka Radwanska defeated Mariya Koryttseva 6-4, 6-3.
Each of the first three days of the French Open has been affected by rain, but Tuesday was the worst yet, with less than three hours of play the entire day. Only 13 matches were completed, five from the men's tournament and eight from the women's.
The French tennis federation has said it might build a retractable roof by 2012.
"We definitely would like to have that," said the 22nd-seeded Mauresmo.
Mauresmo, who has never gotten past the quarterfinals at Roland Garros, was broken in her first two service games and then again while serving for the first set at 5-4.
Leading 6-5 and again serving for the set, the Frenchwoman double-faulted for the fourth of her nine times to give her Ukrainian opponent a break point, but Mauresmo saved it with a backhand winner. She won the set when Savchuk sent a forehand into the net.
In the second set, Savchuk jumped to a 2-0 lead, but Mauresmo got back to 2-2 before the rain. When they came back on court, each player held serve until Savchuk broke Mauresmo while leading 5-4.
Mauresmo dominated the third set by winning four straight games at the start.
"The third set was totally different from the others, that's for sure," Mauresmo said. "In the middle of the second set, after we came back after the break because of the rain, I felt a bit down. I probably ill-managed my food intake."
The fourth-seeded Kuznetsova defeated Aiko Nakamura of Japan 6-2, 6-3 before the rain interrupted play.
"I was first. I was lucky with that," said Kuznetsova, the 2004 U.S. Open champion and twice a Grand Slam runner-up. "I get in my match, so I have the rest of the day to relax, to take it easy, and tomorrow I practice and get prepared for my next match."
Despite the soggy weather, Kuznetsova didn't appear to have any problems against the 71st-ranked Nakamura.
"I hit the ball big, but I also do hit heavy," Kuznetsova said. "And when the ball with these conditions was very heavy, it was very comfortable for me.
"I was a bit weaker in the second set," the Russian added. "First few games I didn't play very well."
Nakamura has never reached the second round in four appearances at the French Open.
"I don't have pressure," Nakamura said, referring to her match with Kuznetsova. "A little bit nervous."
Justine Henin, who won seven Grand Slam titles before abruptly retiring this month at age 25, said on the eve of the tournament that she'd like to see Kuznetsova win and even gave her friend a pep talk to that effect.
"I had a short chat with her," Kuznetsova recounted. "She said, 'Come on, maybe it can be your year.' Yeah, she cheered me up, so it's good. So I thank her for that."
Safina, seeded 13th, defeated Kateryna Bondarenko of Ukraine 6-1, 6-3.
"I'm really happy that I could go through in two sets, especially before the rain started," said Safina, who missed the Italian Open with a back injury.
Safina, who was the last person to beat Henin before the top-ranked Belgian retired, said she still had pain in her back.
"This is my weakest part of my body: the back," said Safina, the younger sister of two-time Grand Slam champion Marat Safin. "So I really had to go home and I had to take some care of it."