This clash stood out for the pedigree of both players on clay and it was seventh seed Stephens, the runner-up to Simona Halep 12 months ago, who was the steadier to claim a 6-4 6-3 victory.
"I thought it was a good match," said the American. "I thought we both played really well. It was very competitive. I think she's been playing some good tennis, and I think so have I.
— Roland-Garros (@rolandgarros) June 2, 2019
"Clay is very neutral. She plays really well on this surface. And obviously since she is such an intense and aggressive player, you kind of have to match her energy.
"I think you can't get passive when you're playing her, so I knew from the very first point I was going to have to step it up and hit my shots. And that's what I did was try to match her the whole way through."
Stephens, who is the highest-ranked player left apart from Halep, will face 26th seed Johanna Konta in the last eight after she defeated Donna Vekic 6-2 6-4 to continue her impressive run at the tournament.
Vondrousova has been in superb form, winning 25 of her last 30 matches, and the 19-year-old was utterly dominant in a 6-2 6-0 victory over 12th seed Anastasija Sevastova.
"It's amazing," said Vondrousova. "I played a great match. She's a tough player, so I'm just happy to be through and happy with my game. You can't expect you're going play quarter-finals in a grand slam, but I played some tough matches."
No woman has won more matches on clay this season than Martic, who had reached the fourth round at a grand slam four times previously without making it further.
She admitted that played on her mind during a 5-7 6-2 6-4 victory over Kaia Kanepi, saying: "It was tough mentally for me today, especially because I was stopped four times before in the fourth round, so it really felt like I was doubting whether I was ever going to get that fourth round (win).
"Two years ago I was two points away. It didn't happen. Today it played a role in my mind. It was not always easy to focus on my game, but I just tried to fight as hard as I could, and thank God, I finally made it."