The Italian, the top seed for the tournament in Paris, avenged her loss to Suarez Navarro at the recent Australian Open with victory in a match that lasted just over three hours.
Errani took a see-saw first set with a break in the 11th game, but the second set saw Suarez Navarro fight back to level. It then appeared the Spaniard would repeat her first-round win in Melbourne when she led 5-4 in the decider.
Yet Errani managed to fend off a match point when serving in the 10th game before claiming the crucial break of serve in the next.
Her next opponent will be Kiki Bertens of the Netherlands, who saw off the 18th-ranked Lucie Safarova 6-1 7-5 to reach her first semi-final at a premier WTA Tour event.
The 21-year-old, who only made the main draw as a lucky loser following the illness-enforced withdrawal of Czech Lucie Hradecka, broke sixth seed Safarova three times in a one-sided opening set.
"It's been an amazing week. My tennis is going very well, I'm very happy with that," said Bertens. "I think I played well today. I was concentrated from the first point and I think that was a good thing."