"I think it's very important, and I'm really happy," said Kuznetsova.
"I was lucky I wasn't cancelled. I got in my match, so I have the rest of the day to relax, to take it easy, and tomorrow I will practise and get prepared for my next match."
Kuznetsova's hopes of winning a second Grand Slam title were boosted when defending champion Justine Henin abruptly retired before the tournament.
"I had a short chat with her when she was here. She said, 'Come on, maybe it can be your year'," said Kuznetsova, 22.
"She cheered me up so I thank her for that. I had a very good relationship with her, respected her when she played. Now I respect her with her achievements.
"She was one of the best athletes, and it's really sad she is leaving."
Mauresmo has dropped out of the world's top five following a dreadful run of form.
The 28-year-old, who has never got beyond the last eight in Paris, edged a low-quality first set before the players were forced to take cover again.
On their return, she dropped the second set against Ukrainian Savchuk but recovered her poise to race through the decider in 34 minutes.
"The problem was my serve because I have a minor injury," said Mauresmo, who won both the Australian Open and Wimbledon in 2006.
Polish teenager Agnieszka Radwanska, the 14th seed, booked a second-round meeting with Croat qualifier Jelena Pandzic with a routine 6-4 6-3 win over Mariya Koryttseva.
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