Williams, the champion at Roland Garros in 2002 and 2013, beat Czech Andrea Hlavackova 6-2 6-3.
But Canadian sixth seed Bouchard lost 6-4 6-4 to France's Kristina Mladenovic, the world number 44, her eighth defeat in nine matches.
Men's 10th seed Grigor Dimitrov lost 7-6 (9-7) 6-2 6-3 to American Jack Sock.
Men's title favourites Novak Djokovic and Rafael Nadal won in straight sets.
Seventh seed David Ferrer saw off Lukas Lacko 6-1 6-3 6-1 to record the 300th clay-court win of his career, while women's fourth seed Petra Kvitova survived a scare before beating New Zealand's Marina Erakovic 6-4 3-6 6-4.
Bouchard hits a low point
Eugenie Bouchard was unable to explain the spectacular loss of form that has seen her win just three matches since reaching the Australian Open quarter-finals in January.
The 21-year-old now heads into the grass-court season knowing she has lots of ranking points to defend having reached last year's Wimbledon final.
"I feel like I have been trying to work on what's been going wrong, and I feel like I have been making progress, so to still have matches like this is actually disappointing," said the Canadian.
"But at the same time it's just a tennis match and, you know, I need to not worry too much. Life is still good. Everyone has highs and lows in their career. This is a little bit of a low point for me."
Rafa requests choice of umpire
Rafael Nadal has confirmed a story that appeared in last week's Telegraph saying he had requested not to have Brazilian Carlos Bernardes umpire his matches.
The Spaniard fell out with Bernardes at the Rio Open in February, with the player claiming he was not allowed to change his shorts after putting them on the wrong way around without incurring a time violation.
"It was my request, and the ATP talking about it," said Nadal. "I asked if it's possible, but nothing personal against him, 100%. I respect him as an umpire, I respect him as a person, and I consider him a good person more than that.
"I would love to have Bernardes on the court again. It will happen, but I think for both of us it is better to have a break. We had some problems. For me he was not enough respectful with me in Rio de Janeiro. That was my feeling when I put my shorts the other way."
Britwatch
The three remaining Britons in the singles draw will not be in action again until Thursday as the schedule evens out following the three-day first round in Paris.
Andy Murray has at least learned that he will face Joao Sousa after the Portuguese beat Canada's Vasek Pospisil 6-3 7-6 (7-5) 6-1 on Tuesday.
The Briton has won all five of his previous meetings with Sousa in straight sets, although they have yet to play on clay.
Heather Watson will take an impressive 4-0 record in against Sloane Stephens, while Kyle Edmund trails fellow 20-year-old Nick Kyrgios 2-0, the first of which was the Canadian Open junior semi-finals in 2012.
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