Medvedev had enjoyed his best run in Paris 12 months ago when he reached the quarter-finals.
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 quarter-finals 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 quarter-finals 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 quarter-finals, 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 break points 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.
The 23-year-old is also into his first ever Grand Slam quarter-final as he continues an impressive season which has seen him win two clay-court titles.
Swiatek survived a scare at the hands of injured Chinese teenager Zheng Qinwen, dropping the first set before going on to make the quarter-finals.
Swiatek prevailed 6-7 (5/7), 6-0, 6-2 to equal the third best winning streak this century of 32 matches set by Justine Henin 14 years ago.
"She played amazing tennis," said Swiatek. "I am proud to be still in the tournament."
In an 82-minute opening set, 74th-ranked Zheng saved five set points, had two of her own and then clawed her way back from 2/5 down in the tiebreak to stun the top seed.
As the 2020 Roland Garros champion's streak looked in peril, Zheng required a medical timeout at 0-3 in the second set for a leg injury.
Zheng, who had defeated 2018 champion Simona Halep on her way to the fourth round, returned with her right thigh heavily strapped and quickly dropped the second set.
Swiatek carved out a double break in the decider against her tiring opponent, whose injury contributed to her 46 unforced errors, and will face US 11th seed Jessica Pegula for a place in the semi-finals.
Pegula downed Romania's Irina-Camelia Begu 4-6, 6-2, 6-3 to add a last-eight place in Paris to similar runs at the Australian Open in 2021 and 2022.
Daria Kasatkina and compatriot Veronika Kudermetova ensured there will be a Russian in the semi-finals after they set-up a last-eight clash.
Kasatkina, seeded 20th, took advantage of Italian Camila Giorgi's 37 unforced errors to win 6-2, 6-2.
Kudermetova reached her first Grand Slam quarter-final as the world number 29 came back from a set down to defeat 2018 semi-finalist Madison Keys of the United States, 1-6, 6-3, 6-1.