Barbora Krejcikova, the 2021 champion, lost her opening match at the French Open for the second straight year. The 13th-seeded Czech exited the tournament after losing to Lesia Tsurenko of Ukraine 6-2, 6-4.
Gauff looked uncomfortable early on, with her troublesome forehand frequently letting her down.
World No. 71 Masarova, a former French Open junior champion, took full advantage to dominate the opening set and looked on course for an upset victory.
But Gauff, 19, settled down on a sunlit Court Suzanne Lenglen and reeled off seven games in a row from 1-1 in the second set to take command.
With her confidence restored, Gauff made no mistake as she moved 5-2 ahead in the decider and wrapped up victory with a hold to love as Masarova sent an attempted lob long.
Entering this week, Gauff said, she thought about what her response should be when asked whether last year's run to the final in Paris should make her feel more pressure.
"At first, I was like, 'I have to match last year's result, blah, blah, blah, or do better.' [But] that was last year," she said Tuesday. "I'm like, 'It's over. What can I do about it?'" she said, before relating that same sort of mindset to Tuesday's match.
"With the first set, it's over," Gauff said. "Like, 'What can I do about it?' You have the choice to dwell on it or reset, and I chose to reset."
Swiatek had beaten Busca in under an hour, losing just one game, at this year's Australian Open but had a less than perfect run-up to Paris, retiring from her Italian Open quarterfinal after suffering a thigh injury two weeks ago.
The 21-year-old triple Grand Slam winner was broken twice in her first three service games as a determined Bucsa, ranked 70th in the world, mixed it up and initially succeeded in throwing her opponent off balance. But the top seed heeded that wake-up call toward the end of the set and broke back to seal it.
The Pole, who began her 61st week at the top of the WTA rankings and is the favorite in Paris, snatched another break at the start of the second set with a thundering crosscourt forehand winner and never looked back.
"I just felt more tense, for sure," Swiatek said. "But I managed to turn that around and not think about all the stuff that is usually bothering when you play your first match."
Jabeur, a crowd favorite in Paris, smiled and expressed relief in not repeating last year's mistake, when she lost to Magda Linette of Poland.
"I'm very happy to win my first match on Philippe Chatrier -- because I've never won here," Jabeur said on court about the clay-court tournament's main stadium.
Now she can focus on trying to win her first major. She was runner-up at Wimbledon and the US Open last year.
Jabeur, 28, of Tunisia, has also battled injuries this season. She had knee surgery after the Australian Open and was then sidelined with a calf injury. She had stopped playing against top-ranked Iga Swiatek at the clay-court tournament in Stuttgart, Germany, in late April and then pulled out of the Madrid Open.
"It was a very difficult period for me after Stuttgart," said Jabeur, adding that she's beginning to find her rhythm.
Jabeur struck 27 winners to Bronzetti's seven, though with 24 unforced errors she'll have room to improve.
"I feel 100% fit. I need to have more matches under my belt to gain confidence," she said.
Bianca Andreescu, who won the 2019 US Open but has been as far as the fourth round at a Slam just once since then and is now ranked 42nd, took out two-time Australian Open champion Victoria Azarenka 2-6, 6-3, 6-4.
Also Tuesday, 16-year-old Russian Mirra Andreeva had a memorable Grand Slam debut by dominating Alison Riske-Amritraj 6-2, 6-1. Andreeva's older sister -- 18-year-old Erika -- lost to Emma Navarro 2-6, 6-3, 4-6 later in the day.
Reuters and The Associated Press contributed to this report.