Two-time Grand Slam champion Victoria Azarenka of Belarus also reached the third round Wednesday in Paris, but US Open champion Emma Raducanu and No. 4 Maria Sakkari went out in the second round. American Sloane Stephens came from behind to defeat No. 26 Sorana Cirstea 3-6, 6-2, 6-0 and advance.
Gauff twice fell behind by two games in the second set, but the 18-year-old broke back both times and then forced three errors from the 28-year-old van Uytvanck in the tiebreaker. Gauff is trying to improve on her quarterfinal run at the French Open last year -- her best result at a major.
Gauff is the youngest player remaining in the draw and celebrated her high school graduation this week with pictures at the Eiffel Tower. Van Uytvanck's best result at a Grand Slam tournament was reaching the quarterfinals at Roland Garros in 2015.
Raducanu, 19, lost to Aliaksandra Sasnovich of Belarus 3-6, 6-1, 6-1, ending her tournament debut after failing to convert on five break-point chances at 1-1 in the third set.
Raducanu's loss means all four reigning Grand Slam champions have been eliminated by the second round -- the first time that has happened since the 2014 French Open, according to the WTA. 2021 French Open champion Barbora Krejcikova was knocked out in the first round, and reigning Wimbledon and Australian Open champion Ash Barty retired in March.
Sasnovich won 12 of 14 service points in the second set en route to her 16th career victory over a top-20 player. She advanced to the third round at Roland Garros for the first time in her seventh appearance.
The 28-year-old had also beaten Raducanu at Indian Wells last year in the British player's first match after winning the title at Flushing Meadows.
No. 15 Azarenka had only 13 unforced errors to Andrea Petkovic's 42 in a 6-1, 7-6 (3) win. Azarenka has twice won the Australian Open and was a semifinalist at Roland Garros in 2013.
Petkovic, 34, reached the French Open semifinals in 2014 but has not been beyond the third round at any Grand Slam since then. She is a five-time clay-court champion on the tour.
Sakkari, who reached the French Open semifinals last year, lost two tiebreakers in falling 7-6 (5), 7-6 (4) to Karolina Muchova. She is the second top-four seed, along with Krejcikova, to go out in the first two rounds.
Czech player Marie Bouzkova withdrew from the French Open before her second-round match, saying she tested positive for COVID-19. She is the first singles player to pull out of Roland Garros because of the coronavirus.
Bouzkova was supposed to face No. 31 Elise Mertens, who advanced to the third round. Bouzkova, 23, has never made it past the second round of a Grand Slam tournament in 13 appearances, including 10 losses in the first round.
She tweeted that she hopes she "will start to feel better soon and get back on court."
The Associated Press contributed to this report.