The top-seeded Williams dictated most points and wound up with 30 more winners -- but also 24 more unforced errors -- than Lim.
"I was a little nervous, like I always am in my first round," Serena said. "It's kind of always hard for me to shake those nerves and go from there."
Third-seeded Agnieszka Radwanska also won her first-round match in straight sets earlier Sunday.
Williams is a 17-time major champion who was facing a woman making her Grand Slam debut.
Still, there was a bit of shakiness from the 32-year-old American as she served out the final game, erasing four break points and needing five match points to close things. Williams saved all seven break points she faced in the match.
She improved to 54-1 in first-round matches at majors. The only loss came at the 2012 French Open against 111th-ranked Virginie Razzano.
Radwanska and Zhang Shuai traded breaks in the opening seven games of their match, but the third-seeded Pole finally held and won the final nine games to become the first player to reach the second round at the French Open, winning 6-3, 6-0.
"So many breaks in a row," said Radwanska, who blamed her struggles on the wet court on an overcast day at Roland Garros. "I don't know if there is really any explanation why there was eight breaks in a row, but I guess it happens sometimes."
Radwanska reached the Wimbledon final in 2012, but her best result at Roland Garros was a spot in the quarterfinals last year.
"Clay is not really my favorite surface, but I'm trying," Radwanska said. "Last year I did my first quarterfinal, so hopefully this year I can do even better."
Seven-time Grand Slam champion Venus Williams, No. 14 Carla Suarez Navarro of Spain and No. 31 Daniela Hantuchova of Slovakia also advanced, along with No. 8 Angelique Kerber. The only seeded player to lose was No. 25 Kaia Kanepi of Estonia. Francesca Schiavone, the French Open champion in 2010 and runner-up in 2011 but now ranked only 53rd, was beaten 6-3, 6-3 by 71st-ranked Ajla Tomljanovic of Croatia.
"I definitely feel like I'm proud of her, but at the same time I want her to keep going and do better and do more and do more," Serena said of her big sister. "And then our relationship, we have just grown older, we are still really close. It's evolved, where it's evolved, I think in a good way."
Serena could meet Venus in the third round if both get through their next matches.
Radwanska started her match with a break to a take a 1-0 lead, but her serve failed her moments later when Shuai was able to even the score. The same happened over the next four games.
But after a fourth break, Radwanska finally held to make it 5-3.
"When I just broke myself in the end of the first set I think was more confident, and then I think I start to play much better," Radwanska said.
The Associated Press contributed to this report.