Gauff made a flurry of unforced errors in her opening match and struggled with her serve in a tearful second-round win, but the 2023 champion kept her cool to subdue the 28th seed.
"It's been an emotional week," Gauff said. "I think I needed those tough moments to be able to move forward. I was putting so much pressure on myself."
After trading early breaks, Gauff broke the Pole in the eighth game and pumped her fist in celebration as Frech sent a shot out on set point.
She outlasted Frech in a 20-shot rally to get the break in the third game of the second set and broke her to love in the fifth as the unseeded Pole made a number of unforced errors.
The American was firing on all cylinders from the baseline in the final game and gave credit to her devoted fans after sealing victory.
"Today showed I was really having fun out there," said Gauff, who booked her fourth straight trip to the Flushing Meadows round of 16. "The support means a lot."
She will next play Australian 15th seed Daria Kasatkina or twice former champion Naomi Osaka of Japan.