The 25-year-old Romanian had never beaten the 21st-seeded Suarez Navarro in four previous matches on clay. Halep improved her career record against Suarez Navarro to 7-5.
"I was expecting a very tough game," said Halep, the 2014 French Open runner-up. "I managed to push her back."
Halep struck 19 winners in 60 minutes of play on Court Philippe Chatrier to return to the last eight in Paris for the second time in her career.
"I think I played the best match here in Roland Garros against Carla, which always was a tough match," Halep said afterwards. "On clay I never won against her, so it was a really good match. I think I played exactly what I had to.
"I'm ready to go into this second week. I'm happy to be in the quarterfinals again, so let's see. Next opponent, it's a big challenge for me. I will give my best."
Halep will next face fifth-seeded Ukrainian Elina Svitolina, who rallied to beat Croatian qualifier Petra Martic 4-6, 6-3, 7-5 on Court Suzanne Lenglen.
Svitolina reeled off five straight games from 2-5 down in the final set for a hard-fought victory against the 290th-ranked Martic.
"It was amazing tennis from Petra today, and she was really striking the ball well," Svitolina said. "I was really happy to come back in the second set. Thanks to the crowd for cheering me on. They gave me the energy to fight."