US Open champion Medvedev was beaten by 20th seed Marin Cilic 6-2, 6-3, 6-2 in just 1hr 45 min on Court Philippe Chatrier.
Tsitsipas, the 2021 runner-up to Novak Djokovic, was defeated by Danish teenager Holger Rune 7-5, 3-6, 6-3, 6-4 earlier in the day.
Cilic will face Russia's Andrey Rublev in his third career Roland Garros quarterfinal.
"It's one of the most beautiful matches of my career, it was fantastic from the first to the last point," said Cilic, adding he is playing his "best tennis in recent weeks".
Medvedev had enjoyed his best run in Paris 12 months ago when he reached the quarterfinals.
He hadn't dropped a set at this year's tournament but was completely outplayed by the 33-year-old Cilic who broke serve five times.
Medvedev was unable to carve out a single break point as the Croatian got the better of the Russian for the first time in four meetings.
Rublev made the quarterfinals for the second time when Italian opponent Jannik Sinner retired with a left knee injury with the Russian 1-6, 6-4, 2-0 ahead.
Rune became the first Danish man to reach the French Open quarterfinals when he shocked Tsitsipas.
Rune, just 19 and ranked 40, swept to a memorable win on the back of 54 winners.
With fellow 19-year-old Carlos Alcaraz already into the quarterfinals, it's the first time two teenagers have made it this far in Paris since Hendrik Dreekman and Andrei Medvedev 28 years ago.
In a tense fourth set, Rune gave up a 5-2 lead and fought off three more breakpoints in the 10th game before securing victory when Tsitsipas hit long.
"I was very nervous and I knew that if I went away from my tactics I would lose," said Rune, who won his maiden ATP title in Munich in the build-up to Paris.
"I told myself just stick to the plan and that gave me a confidence boost. It's so great to still be here."
Tsitsipas said he would be ready for Rune when they meet again.
"I can see something different next time with this opponent. I'm pretty convinced I can do way better," said the 23-year-old.
Rune will next face eighth-seeded Casper Ruud who became the first Norwegian man to reach the last eight with a 6-2, 6-3, 3-6, 6-3 win against Poland's Hubert Hurkacz.