Defending champion Williams failed to reach top form at Court Philippe Chatrier as the second grand slam of the season got under way on Sunday.
The American - a 17-time grand slam champion - made 36 unforced errors but still overcame Alize Lim 6-2 6-1 in just 80 minutes.
Williams admitted she was feeling nervous after the win but the 32-year-old has time to grow into the tournament, and next faces Garbine Muguruza after the Spaniard overcame Grace Min 7-5 7-6 (8-6).
Third seed Radwanska is still searching for a maiden grand slam triumph, and started her latest challenge with a 6-3 6-0 triumph over China's Shuai Zhang.
The match was dominated by breaks of serve, with Zhang failing to hold throughout and Radwanska - who reached the last eight in 2013 - initially struggling to hold as well.
However, the Pole came through in just over an hour and will meet the winner of the clash between Mathilde Johansson and Karolina Pliskova in round two.
Elsewhere, a partisan home crowd almost celebrated an unlikely success as Amandine Hesse - ranked 221 in the world - pushed Yvonne Meusburger all the way before succumbing to a gallant 3-6 6-3 6-4 defeat.
Hesse, a 21-year-old wildcard entrant, was in inspired form to take the opener against the world number 40, but Meusburger soon levelled the match.
It looked as though the Austrian would ease over the line from there when she opened up a 5-3 lead and served for the win.
However, Hesse broke back against the odds to restore hope, only for the favour to be immediately returned, leaving Meusburger breathing a huge sigh of relief.
The two players shared a warm embrace at the net and Meusburger retired to her chair to allow Hesse to soak up the appreciation of an adoring crowd.
Eighth seed Angelique Kerber's previous best performance in Paris came with a run to the quarter-finals in 2012, and she was in no mood to hang around in a 6-3 6-1 triumph of Katarzyna Piter.
Venus Williams, sister of Serena, will also be involved in round two. The 29th seed ousted Belinda Bencic 6-4 6-1, while fellow seeds Daniela Hantuchova and Carla Suarez Navarro defeated Jovana Jaksic and Yuliya Beygelzimer respectively.
There were some shocks on day one, however. Francesca Schiavone, the 2010 champion, was defeated 6-3 6-3 by Ajla Tomljanovic, while Kaia Kanepi – a two-time quarter-finalist - was the first seeded casualty as she went down in three sets to Monica Niculescu.
Among the other first-round victors were France's Claire Feuerstein, Slovakian Anna Schmiedlova and Sweden's Johanna Larsson, the latter comfortably overcoming former world number 10 Maria Kirilenko, who has been plagued by injury this year.
Tsvetana Pironkova and Varvara Lepchenko also progressed to round two, while the match between Lauren Davis and Yaroslava Shvedova was suspended with the score level at one set apiece.