The younger Williams, a 34-year-old four-time Olympic gold medalist, had a slow and frustrating start against Cornet before turning the momentum to win 7-6 (5), 6-2, picking up the pace in the second set after the first ran over an hour long.
"I just needed to relax. I was missing shots by literally centimetres," Williams said, adding that she "tried to add a little more spin" in the second set, after a long series of unforced errors in the first.
Spain's Garbine Muguruza topped Japan's Nao Hibino 6-1 6-1.
Earlier, Czech Republic's Petra Kvitova knocked out former world number one Caroline Wozniacki in the second round of the Rio Olympics on Monday, in another poorly attended tennis match on centre court.
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Twice Wimbledon champion Kvitova overpowered Wozniacki 6-2 6-4 to progress to the third round, where she will face either Russia's Ekaterina Makarova or Slovakia's Anna Karolina Schmiedlova.
"It was a tough draw for me. Petra played well today, really played aggressively and got me on my heels a little bit," said Wozniacki, ranked 53 in the world after an injury-ravaged season.
Wozniacki's participation had been in doubt as injury prevented her from fulfilling the minimum number of Davis and Fed Cup appearances required in a four-year Olympic cycle to qualify for the Games.
But she made it to Rio, where she was Denmark's flagbearer in the opening ceremony.
Germany's Angelique Kerber, ranked second in the world, faced stubborn resistance from Canada's Eugenie Bouchard but won 6-4 6-2.
United States' Madison Keys, seeded seventh in the tournament, also progressed to the third round after a nail-biting 7-5 6-7(4) 7-6(5) win against France's Kristina Mladenovic on court one.
On the men's side, Gilles Muller of Luxembourg beat Jo-Wilfried Tsonga of France 6-4 6-3.
Monday's matches, like most of the other contests in the first three days of the tennis tournament, were played in stadiums that were more than half empty.
Olympics organisers say about 82 percent of all Olympic tickets have been sold out, but for tennis, which boasts some of the world's most recognisable sports stars, attendance has been far below that level.
(Additional reporting by Joshua Schneyer; Editing by Susanna Twidale and Bill Rigby)
