The reigning US Open champion's fightback into the third round took more than two and a quarter hours, coming after a first-round bye.
World number three Gauff was playing in front of a home crowd for the first time since winning her first Grand Slam title six months ago.
Gauff, who turns 20 next Wednesday, broke a WTA record by reaching 50 wins as a teenage at the Masters 1000 level, a category established in 2009.
Gauff trailed her 47th-ranked French opponent 4-0 in the deciding third set, but was able to power back and throw it into a tiebreaker.
She advanced on her third match point as Burel swung wide.
"I'm happy with the mental fight," Gauff said. "It was not my best tennis... but it's how you show up on your worst days, not your best ones, that counts."
Gauff said that after dropping the opening set, she had to change strategy.
"I tried to play more aggressive," she said. "She was giving me a lot of short balls and I wanted to punish her on them.
"In the third set I was missing, but I finally found my game."
Unseeded Emma Raducanu benefitted as Dayana Yastremska retired four games into their match with an unspecified injury, handing the Briton a 4-0 win and a place in the third round.
Raducanu, whose career trajectory has been patchy since winning the 2021 US Open, was on court for a mere 21 minutes.
She now awaits an opponent from a match between second-seeded double Australian Open champion Aryna Sabalenka and American Peyton Stears.
Raducanu will be playing in the third round here for a third consecutive year.
In the ATP draw, Britain's Cameron Norrie advanced to the third round over Italy's Lorenzo Sonego 6-4, 6-4.
Last weekend's Dubai champion, Ugo Humbert, the 14th seed, defeated American Patrick Kypson by the same scoreline.
