The Kazakh withdrew from the Cincinnati tune-up tournament with a leg injury, raising questions over whether she would even compete in New York, but appeared untroubled as she launched her bid for the title and the world number one ranking.
"I call for the physio in Cincinnati. She said the Achilles tendon was not broken, so that was kind of a good thing, but of course I still had the pain," said Rybakina.
"From that, we started my recovery. We didn't know if I was going to be able to play here or not. I'm pretty happy, of course, that I'm playing right now, and we were taking it day by day. I was not playing for the full week, so seven days no tennis and basically not too much gym; I could do movement-wise.
"So it's a great result after 10 days from that kind of injury."
Rybakina, a two-time Grand Slam champion, will next face either Czech Darja Vidmanova or Spain's Jessica Bouzas Maneiro.
Frodin was making her Grand Slam main-draw debut five years after portraying a young Serena Williams in the Oscar-winning film King Richard.
The 17-year-old tennis-playing double for the film showed flashes of brilliance, with 11 aces, after fans waited nearly four hours through a rain delay to watch her on the Grandstand.
But she struggled to trouble the Australian Open champion in her first tour-level match, as Rybakina broke in the sixth game on the way to taking a largely uneventful opening set and maintaining control in the second.
Frodin drew some of the loudest cheers of the match when she saved three match points with aces in the penultimate game, but Rybakina sealed victory with an ace of her own.
"She played really well today, and she has great shots," Rybakina said before pausing as she packed her kit to join the crowd's ovation for Frodin.
"It was not easy, and I'm just happy to be here, playing here."
