One of the most reliable returners in the sport, Czech Kvitova had won both of their previous meetings but was left to rue her missed chances - as she converted only two of the six break points she earned.
Rain forced a 44-minute stoppage early on in the match, with tournament staff scrambling to close the retractable roof on Ashe and mop up the court.
But eighth seed Pegula was unfazed as she broke Kvitova to love in the seventh and ninth games to close out the first set.
Even when Kvitova took a 2-0 lead in the second, Pegula kept her nerve to storm back and win the next six games to wrap up the win.
"I just told myself not to get frustrated," Pegula said after she joined fellow American Coco Gauff (18) in the last eight. "I just really wanted to stay committed to hitting my spots.
"It feels amazing - I'm so happy. I thought I handled my nerves and stuff really well."
"For me, my placement on my serve was very important. She can really hit it so early so out in front and hit her spots so well.
"That makes it really hard to defend. Even if you get back the first ball of a return you're probably going to lose the point."
She next faces world number one Iga Swiatek (21), who needed three sets to tame 108th-ranked German Jule Niemeier (23).
"I just have to play smart...," said Pegula. "I thought I did a great job of that today so I'm going to try and replicate that."
