The 22-year-old Venezuelan-born fourth seed will next play Shelby Rogers of the US for a place in the semifinals.
"At the end it is always very tough. She is a player with a lot of experience and I just had to be there," she said.
"Quarterfinals is a great run and I am looking forward to -improving my result here."
Muguruza is seen as one of the new stars of the women's game after a breakthrough season in 2015 when she was runner-up to Serena Williams at
Wimbledon and made the semis at the WTA Finals.
But in 30-year-old Kuznetsova she was up against one of the wiliest and experienced players on the women's circuit with two Grand Slam titles to her
name - the 2004 US Open and the 2009 French Open.
Both players had early break points on a sparsely filled Philippe Chatrier center court in a match of high quality.
And it was the Spaniard who struck first in the eighth game powering in a fine backhand -return past the Russian to move 5-3 up.
She then served out com-fortably for the set in 46 minutes.
Muguruza kept up the pressure and broke Kuznetsova's serve twice at the start of the second set, although she dropped her own in between.
The Russian pulled back to level at 4-4 Authentic Harold Landry Jersey , but it was Muguruza who rose to the occasion taking the next two games and clinching the win on her fifth match point.
World No.108 Rogers reached her first Grand Slam quarterfinal with a 6-3, 6-4 win over Romanian 25th seed Irina Camelia-Begu.
Rogers is just the ninth woman outside of the top 100 to make the last eight in Paris since 1983.
"It's incredible and I'm -really happy that it didn't start raining," said Rogers who broke down in tears after the match and had to borrow a handkerchief
from television interviewer Marion Bartoli, the former Wimbledon champion.
Rogers has now beaten three seeds in her run at Roland Garros having seen off 17th seed Karolina Pliskova in the first round and 10th seed Petra Kvitova in
the last 32.
In the men's competition Authentic Rashaan Evans Jersey , Eighth seed Milos Raonic was swept aside 6-2, 6-4, 6-4 by Spanish player Albert -Ramos-Vinolas in the fourth round.
The big-serving Canadian could make little impression against the dogged, 28-year-old left-hander who finished the contest with an angled smash on his
third match point. -Ramos-Vinolas reached his first Grand Slam
quarterfinal Authentic Jack Conklin Jersey , having never been past the second round in his 18 appearances in the four majors.
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