Kvitova served for the first set at 6-5, but was broken at love and then lost four straight points in the tiebreaker to trail 4-3. But she capitalized on a couple of errors by the Spaniard and converted her first set point when Suarez Navarro netted a backhand.
In the second set, she broke to lead 4-2, and served out the match at love when Suarez Navarro sent a forehand long.
The 18th-ranked Suarez Navarro was the first Spanish woman since 2001 to reach the fourth round at the All England Club.
All remaining men and women were set to play their fourth-round matches.