Down 2-4, Gauff fought her way back and saved three break points while serving at 4-all before closing out the match in the next game.
Four years ago, Peterson routed a then-14-year-old Gauff at a Challenger tournament in Michigan.
“She whopped me really bad,” Gauff said on court. “I think today was really a mental thing, just staying in the match. I wasn’t playing my best in some moments and I wasn’t serving as well as I’d like to, but I think that my mentality kept me in.”
The sixth-seeded American next plays No. 2 seed Aryna Sabalenka, who beat 16th-seeded Barbora Krejcikova, 6-3, 2-6, 6-4.
No. 3 Jessica Pegula lost to 15th-seed Petra Kvitova 6-2, 3-6, 7-6 (11) after the American blew four match points
“When the match like this, very exhausting,” Kvitova said. “Up and down. Definitely it’s one of the best matches I’ve played.”
Sorana Cirstea took out No. 5 Caroline Garcia 6-4, 4-6, 7-5, No. 7 Maria Sakkari beat 17th-seeded Karolina Pliskova 6-4, 5-7, 6-3, and No. 10 Elena Rybakina defeated Varvara Gracheva 6-3, 6-0.
On the men’s side, defending champion Taylor Fritz moved into the quarters with a 6-4, 6-3 win over Marton Fucsovics.
No. 5 Daniil Medvedev overcame a swollen right ankle and outlasted 12th-seeded Alexander Zverev, 6-7 (5), 7-6 (5), 7-5.
“When I rolled my ankle, I rolled it pretty hard,” he said. ”At the moment I rolled it I was like ‘Okay I’m going to stand up and it’s going to be fine,’ but then I continued to stay on the ground because the pain was only growing. It was definitely a crazy match.”
Medvedev took a medical timeout to have his ankle taped in the second set before going on to notch his 17th consecutive match win and improve to 22-2 this year.
“This one was definitely the toughest one of all,” he said on court.
Zverev added to the drama by saving a match point and breaking for 5-5 in the final set. Medvedev broke right back and went on to win.
He next plays Spain’s Alejandro Davidovch Fokina, who beat Cristian Garin 6-3, 6-4.
No. 10 Cameron Norrie, the 2021 champion, beat sixth-seeded Andrey Rublev, 6-2, 6-4. Norrie improved to 21-3 this year.
Norrie advanced to a quarterfinal against Frances Tiafoe, who beat qualifier Alejandro Tabilo, 6-4, 6-4.