The 2022 French Open finalist beat fellow 18-year-old Montgomery in a rain-affected match in one hour and 23 minutes, to clinch a last-16 clash with Bianca Andreescu.
The match marked Gauff's first game on tour since losing to Caroline Garcia in the US Open quarter-finals last month, although she had to endure an overnight rain delay, resuming on Wednesday up 6-3 3-2.
Gauff only needed 20 minutes on Wednesday to close out the victory, finishing the match with 3-0 aces, winning 76 per cent on her first serve and converting four of seven break points.
Croatian qualifier Donna Vekic continued her strong run with a 6-3 6-2 win over 2022 US Open quarter-finalist Karolina Pliskova. Vekic moved into the third round, having beaten Maria Sakkari in the first round too.
Fourth seed Jessica Pegula eased her way into the third round with a 6-3 6-1 victory over wild card Coco Vandeweghe in only 69 minutes. Pegula can secure her first WTA Finals qualification this week.
Danielle Collins was another American victor, triumphing 6-2 6-4 over Italy's Martina Trevisan in one hour and 27 minutes, reaching her third quarter-final of the year.
Seeded ninth for the WTA International tournament, Japan's Hibino was upstaged 3-6 6-2 6-3 by Jasmine Paolini on Monday.
American eighth seed Li, however, had no such trouble in her 6-3 7-5 victory over Arantxa Rus on the outdoor hard courts.
Coco Vandeweghe – the 2017 Australian Open semi-finalist – suffered another early exit in Mexico.
After being ousted by Mihaela Buzarnescu in the opening round in Guadalajara last week, former world number nine Vandeweghe was eliminated 6-3 6-2 by Canadian teenager Leylah Fernandez.
Fernandez will face either top seed Sloane Stephens or lucky loser Kristina Kucova in the second round.
Anna Karolina Schmiedlova, Nina Stojanovic, Viktoria Kuzmova, Lauren Davis and Tamara Zidansek also progressed from the opening round.
Garcia, the world number 10, entered the contest off back-to-back losses for the first time since March, and Collins made it three in a row as she was just a little too good with both her serving and return game.
Collins won 63 per cent of her service points, with Garcia at 52 per cent, and she ended up securing five breaks in the match.
She will play Martina Trevisan in the second round after the Italian defeated Colombian qualifier Camila Osorio 6-3 6-4.
Neither player had an ace in the match, but the big differentiating factor was Trevisan's ability to win points off her second serve, converting 50 per cent of her chances while Osorio won only one of nine (11 per cent).
The only qualifier of the day to get a win was Louisa Chirico, who beat fellow American Alison Riske-Amritraj 1-6 7-5 7-6 (7-5).
Chirico, the world number 196, will be rewarded for her win with a showdown against world number four Paula Badosa.
Madison Keys had no issues dealing with Australian qualifier Ellen Perez 6-1 6-4 in just over an hour, and Coco Vandeweghe defeated Sofia Kenin 6-1 1-6 6-4.
In the last match of the night, Canada's Bianca Andreescu won a two-hour-and-40-minute battle against Russia's Liudmila Samsonova 7-6 (7-1) 4-6 6-2.
