The fifth-seeded Italian, who lost last year's final to Maria Sharapova, took a medical timeout near the end of the first set after clutching over in pain and holding her stomach. After a few minutes of treatment, she was able to resume playing.
Errani screamed in delight after closing out a contest lasting 2 hours, 29 minutes on her second match point with a backhand drop shot.
"For me to be in quarterfinals is unbelievable," Errani said on court Suzanne Lenglen. "She's an amazing player and it's always tough to play against her. I'm very happy to have won."
She improved to 6-3 overall against the 20th-seeded Suarez Navarro, who reached the last eight on debut at Roland Garros in 2008.
Errani will play fourth-seeded Agnieszka Radwanska of Poland, who beat 2008 champion Ana Ivanovic of Serbia 6-2, 6-4.
"I'm just going to prepare and do everything in my power to be the first time in the semifinal here," Radwanska said. "For sure it's going to be a lot of good rallies."
Radwanska leads Errani 6-1 overall and 3-1 on clay.
At deuce in the 11th game, Errani doubled up several times and held her stomach before sitting on a chair as a doctor and a physical trainer came onto the court.
Taking the timeout, she lay on her back with towels draped over her thighs as the physio worked on her stomach and the area near her sternum.
Errani explained afterward that she felt a stabbing pain "like a knife" under her ribs.
"Every time I was breathing (it) was very painful in this (spot)," she said. "I know it wasn't muscular, but I couldn't (stand) up. So I called the physio and she told me that I had the diaphragm totally blocked, and she tried to release it a bit. After (it) was much better."
She got back to her feet but was broken for the second time, and Suarez Navarro clinched the first set on serve with a backhand winner down the line.
"I think that from a tactical perspective I approached the match very well," Suarez Navarro said. "Things were pretty clear in my mind, but we all know full well that there is a gap between what you think and what actually happens."