After taking the first set, the sixth-seeded American began to struggle with her forehand in the second and an opportunistic Peterson began to apply pressure by coming into the net.
But after suffering an early break in the decider, Gauff stepped up her defence, fending off three break points at 4-4 and breaking Peterson for a fourth time on match point to book her spot in the last eight.
Gauff, who turned 19 on Monday, was serenaded with a rendition of "Happy Birthday" after the match by the partisan crowd in the Southern California desert.
"Today, it was just a mental thing, staying in the match," Gauff said.
"I wasn't playing my best in some moments and wasn't serving as well as I'd like to, but I think my mentality kept me in today."
Next up for Gauff is a meeting with either Belarusian second seed Aryna Sabalenka (24) or Czech 16th seed Barbora Krejcikova (27), who play later on Tuesday.
Elsewhere, seventh seed Maria Sakkari (27) outlasted big-serving Czech Karolina Pliskova (30) 6-4 5-7 6-3 in a two-hour and 45-minute marathon to reach the quarters.
With the win the Greek keeps alive her hopes of capturing a first Indian Wells crown after making the final last year.
Sakkari will next face either third-seeded American Jessica Pegula (29) or Czech 15th-seed Petra Kvitova (33) for a semi-final spot.