Seventh seed Stephens struggled with her serve but she was more consistent on Court Philippe Chatrier and set up a meeting with Britain's Johanna Konta in the last eight.
The 2017 U.S. Open winner is now the clear favourite in her half of the draw after the exits of world number two Karolina Pliskova and one of the pre-tournament favourites Kiki Bertens of the Netherlands.
"I think Mugu is one of the most intense players out on tour. She has a lot of, like, good, strong energy. You can feel it from the other side of the court," Stephens said.
"And I think sometimes if you get too passive, that
kind of takes over you and your game, like, what you're trying to do.
"So like I said, you just have to match that. You have to stay intense. You have to move your feet. You got to do all the things that you know how to do to combat that energy."
Stephens shook off a wobbly start by breaking back for 3-3 and she broke 2016 champion Muguruza's serve again to move 5-3 up.
The American's serve was still far from perfect, however, and she surrendered the extra break right away.
She bagged the opening set, however, on her opponent's equally inconsistent serve.
Another break in the second set allowed her to serve for the match and she wrapped it up on her fifth match point when 19th seed Muguruza returned a second serve poorly.
She is the first American of the singles draw in the last eight and could be followed by Madison Keys, Sofia Kenin and Amanda Anisimova who are all in fourth-round action on Monday.
(Reporting by Julien Pretot; Editing by Toby Davis and Ed Osmond)
