Osaka, who took a 16-month hiatus from tennis in September 2022 to start a family, was inconsistent in the first match on Court Philippe Chatrier of the tournament but secured a 6-1, 4-6, 7-5 win.
"It feels really nice to be back and I'm just really grateful to be here in front of everybody," said Osaka.
"I think there were moments when I played really well."
She will next face her fellow four-time Grand Slam champion Swiatek if the Pole sees off Leolia Jeanjean on Monday.
The pair have only played each other twice before, with their last meeting a Swiatek victory in the 2022 Miami Open final.
Sunday was Osaka's first match win at a Grand Slam event since the 2022 Australian Open, after losing in the French Open and US Open opening rounds that year and at Melbourne Park earlier this season.
The Japanese star has never reached the second week at Roland Garros, with all of her major titles coming on the hard courts of Melbourne and New York.
She also withdrew from the tournament in 2021 to protect her mental health after being fined for opting out of mandatory media commitments.
Osaka, currently world number 134 as she steadily makes her way back up the rankings, was in fine form early against Bronzetti, wrapping up a dominant first set after just 26 minutes with an ace.
She did not face a break point until she dropped serve in the 10th game of the second set as the match went into a decider.
The former world number one quickly retook control, reeling off the first four games of the third set.
But Osaka missed a break point for a 5-0 lead and then fell apart as Bronzetti powered back, recovering the two breaks and moving 5-4 ahead.
The 26-year-old Osaka managed to gather herself, though, breaking again in the 11th game before taking her first match point when Bronzetti netted a forehand.
