The Spaniard, winner in 2016 and seeded third this year, took only 62 minutes to see off Australia’s Stosur, the former US Open winner, 6-0, 6-2.
Muguruza seemed unfazed by the favourite’s tag. “I guess they say that because I played a good match. It’s the normal thing,” she said.
“But I believe I can hold a trophy not only here, I believe I can do it in every tournament I play.
“I’m pleased with my performance. It’s never an easy match against a grand slam champion. She played a final here. I’m pleased because I felt in control in the match.”
Two-time Wimbledon winner Petra Kvitova was a third-round casualty, the eighth seed going down in two tie-breaks to Anett Kontaveit.
Top seed Simona Halep came through 7-5, 6-0 against Andrea Petkovic of Germany. Halep, of Romania, next plays 16th-seeded Elise Mertens of Belgium. Mertens won 6-3, 6-1 against No24 seed Daria Gavrilova.
Two-time Grand Slam winner Angelique Kerber is through to the fourth round after winning 7-6 (4), 7-6 (4) against No 18 seed Kiki Bertens.
The 12th-seeded German has reached finals at every Grand Slam tournament except the French Open. Her best performance at Roland Garros was a quarter-final appearance in 2012.
She next plays Caroline Garcia of France who beat Irina-Camelia Begu.