French Open: Sunday 25 May to Sunday 8 June
Coverage: BBC TV, BBCi, Radio 5 Live and the BBC Sport website.
Ivanovic is the second seed at Roland Garros
Last year's finalist Ana Ivanovic went through to the second round of the French Open despite an inconsistent display on the opening day in Paris.
The Serb second seed beat Sweden's Sofia Arvidsson 6-2 7-5.
"Today was the first match of the tournament so I was a little bit nervous," said Ivanovic. "She gave me a tough match."
Serena Williams, the 2002 champion, crushed fellow American Ashley Harkleroad 6-2 6-1.
Williams, the fifth seed, trailed 2-0 in the first set before roaring back to claim victory, sealing it with her 10th ace.
"I'm very happy with the result, although I could have played better, but in these first rounds you just want to get them over with," said Williams.
"I don't feel I have any pressure, at least I don't put pressure on myself. And I think I have a good chance here."
Nicole Vaidisova, the 15th seed, became the first high-profile casualty when she was beaten by Iveta Benesova.
Vaidisova, who has been working with Tim Henman's former coach David Felgate, lost 7-6 (7-2) 6-1, her sixth straight defeat.
Ivanovic, 20, struggled with her serve on Sunday and had to come back from 3-1 down to claim the second set.
"This year I'm the world number two and obviously have more expectations for myself, so that's something I'm still learning how to deal with," said Ivanovic, who produced 30 unforced errors.
The Serb is likely to be given a sterner test in her second-round match against Lucie Safarova, who crushed German Sandra Kloesel 6-1 6-1.