Gauff, 15, beat Babos 6-2, 4-6, 6-4 on Thursday at USTA Billie Jean King National Tennis Center in Flushing, N.Y., to advance to the third round.
No. 1 seed Naomi Osaka reached the third round at the 2019 U.S. Open with a 6-2, 6-4 win over Magda Linette.
Osaka, with NFL free-agent quarterback Colin Kaepernick and former Los Angeles Lakers star Kobe Bryant watching on, defeated Linette earlier in the day to set the stage for a highly anticipated showdown with Gauff, who had a magical run at Wimbledon this year.
Osaka, the defending women's U.S. Open champion, fell behind 3-0 in the second set but reeled of five straight games to seal the victory.
In other women's action, unseeded American Taylor Townsend, 23, stunned Wimbledon champion and No. 4 seed Simona Halep to move on.
Halep grabbed the opening set before losing two straight against Townsend, who claimed a 2-6, 6-3, 7-6 win. Halep had a match point before Townsend won in a thrilling tiebreak.
Yulia Putintseva picked up an impressive win over Aryna Sabalenka, upsetting the No. 9 seed 6-3, 7-6 (3) to reach the third round for the first time.
No. 7 seed Kiki Bertens edged Anastasia Pavlyuchenkova 7-5, 6-4. Great Britain's Johanna Konta, the tournament's No. 16 seed, dominated Margarita Gasparyan 6-1, 6-0 to advance.
In the men's draw, No. 6 seed Alexander Zverev punched his ticket to the third round at Flushing Meadows with a five-set win over American Frances Tiafoe.
Zverev edged Tiafoe 6-3, 3-6, 6-2, 2-6, 6-3 to push his record to 5-0 in five-set matches this year.
Fifth-seeded Daniil Medvedev eliminated Hugo Dellien with a 6-3, 7-5, 5-7, 6-3 victory. It marked his 46th tour-level match win this season, the most by any player on the men's tour.
No. 14 seed John Isner advanced to the third round with a straight-sets win against Germany's Jan-Lennard Struff. No. 2 seed Rafael Nadal received a walkover into round three after Thanasi Kokkinakis withdrew.