After a back-and-forth first set, the No. 4 seed Spaniard broke the 27th-seeded Russian's serve three times in the second on Court Philippe-Chatrier in Paris.
"I'm happy I went out there without any kind of fear," Muguruza said afterward . "I wanted the match. I tried to just do what I had to do and keep that during the whole match.
"I was not thinking a lot because it's, you know, a lot of thinking in this tournament for me in the matches and stuff, so I try to go out there and block everything and just think about what I have to do to win. The more matches I play -- and tough victories -- I think it gives you self-confidence, a successful feeling out there."
Muguruza will next face 13th-seeded Kristina Mladenovic of France, who rallied to beat American Shelby Rogers 7-5, 4-6, 8-6.
Mladenovic, down 5-3 down in the last set before breaking Rogers twice, secured the victory on her first match point with a forehand winner on Court Suzanne-Lenglen.
Mladenovic will reach the second week of her home Grand Slam for the first time.
"I went through so much emotion in this match, I don't know what to say," Mladenovic said on-court to the fans after the match in two hours and 46 minutes. "I don't have words to describe the love I have for you and what you bring me."
Rogers, who was a surprise quarterfinalist in Paris a year ago, saw Mladenovic win 25 of the last 33 points to pull off the comeback.
"She didn't really miss a ball. I think she elevated her game," Rogers said of Mladenovic's turnaround. "Big round of applause for her. She played very well. The French fans are awesome. They love their French players, and it's a really cool atmosphere, I guess, try to pretend they are cheering for me. They have the wave going and everything. It's nice to see the fans of tennis like that.
"I think it was a lot of positives to take from this tournament. Even today, you know, let the first set get away, then refocused, played really well. I'm proud of myself for doing that. Coming back in the second and fighting until the end."
No. 23 seed Samantha Stosur of Australian notched her eighth consecutive win with a 6-2, 6-2 victory over American qualifier Bethanie Mattek-Sands, moving into the second week of the French Open for the sixth time in her career.
Stosur will next meet 19-year-old Jelena Ostapenko of Latvia, who beat Lesya Tsurenko of Ukraine 6-1, 6-4.