Gauff raced into an early 3-0 lead in the first set, before she double-faulted on break point in the fifth game as Frech clawed her way back to 3-3.
That wobble could have turned into a nosedive, but Gauff composed herself and won the next three games to wrap up the set in 43 minutes.
After making 79% of her first serves in the opening set, Gauff continued to build confidence in the second.
The two-time Grand Slam champion hit serves of 111mph and 113mph as she continued to improve her speeds from earlier in the tournament, and made just two double faults after making 18 in her first and second-round matches combined.
Gauff broke twice more in the second set to build a commanding lead, before breaking for a third time to secure her place in the next round.
She will face either 15th seed Daria Kasatkina or two-time US Open champion Naomi Osaka in the last 16.
Meanwhile, 11th seed Karolina Muchova came from a set down to defeat Linda Noskova 6-7 (5-7) 6-4 6-2.
Back-to-back breaks at the end of the first set resulted in Noskova winning a tie-break, but mistakes cost the Czech player as she committed 15 double faults.
Muchova overpowered her compatriot to set up a last-16 meeting with either Diane Parry or 27th seed Marta Kostyuk.
