Fruhvirtova, matched Gauff in the early stages but once the American grabbed a 3-2 lead there was no looking back.
Gauff raced to 5-0 lead in the second set, and while the 19-year-old faced two break points serving for the match she saved both to win her first career meeting with the Czech.
"(Fruhvirtova) started off playing really well, and I was able to raise my level," Gauff said. "I thought I served well and did better on the returns, and I think I played a great match today."
Emma Raducanu, playing her first tournament following an eight-month injury layoff, was unable to claim a place in the last eight after the former Flushing Meadows champion lost 6-7(5), 7-6(3), 6-1 to second seed Elina Svitolina.
Briton Raducanu won the first set of her match and led 3-1 in the second set tie-break before Svitolina fought back to force the decider.
The Ukrainian upped her game in the third set to take the match and set up a clash with Marie Bouzkova.
