Fruhvirtova (16), 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 (21), 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.
At the Brisbane International, another tune-up event for the Melbourne Park Grand Slam, which runs from January 14-28, former French Open champion Jelena Ostapenko beat Karolina Pliskova 6-2, 4-6, 6-3 to reach the quarters.
Latvian Ostapenko, who will meet former world number one Victoria Azarenka for a place in the last four, raced through the first set but looked frustrated as Pliskova fought back to claim the second.
"I get really mad when I miss some important points and I get really angry at myself," Ostapenko said.
"I'm a fighter and I really want to play kind of perfect. But I know nothing is always perfect. It’s just the way I am."
Two-time champion Azarenka beat France's Clara Burel 7-5 6-2, the 34-year-old Belarusian powering to a 4-1 lead in the opener before Burel drew level.
"I was trying to be aggressive, come to the net, but I felt like she was really on top of me," Azarenka said. "She did some amazing passing shots, some counter-attack balls."
Elsewhere, top seed Aryna Sabalenka made short work of Lin Zhu, world number four Elena Rybakina thrashed Elise Mertens 6-1, 6-0 and talented youngster Mirra Andreeva eased past Arina Rodionova 6-1, 6-1.
