Rune, just 19 and ranked 40, swept to a memorable 7-5, 3-6, 6-3, 6-4 win on the back of 54 winners against the 2021 runner-up from Greece.
He will face Norway's eighth seed Casper Ruud for a place in the last-four.
With fellow 19-year-old Carlos Alcaraz already into the quarter-finals, it is 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 he secured 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 next time.
"He is a very emotional player, he can play great, he absolutely deserves this victory, played better, faced crucial tough moments better," said the 23-year-old.
"But I can see something different next time with this opponent. I'm pretty convinced I can do way better."
Ruud 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.
- Ruud in last-eight -
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.
"It's been a goal of mine to reach the quarter-finals of a Grand Slam," said Ruud.
Ruud will start favourite in Wednesday's match, having won all three of his previous ATP meetings with Rune, all on clay and without dropping a set.
One of those wins came at Monte Carlo as recently as last month.
In the women's draw, Kasatkina said she is looking forward to sharing another "good memory" with Kudermetova when the Russian duo face each other in the quarter-finals.
Kasatkina, seeded 20th, took advantage of Italian Camila Giorgi's 37 unforced errors to win their last-16 tie, 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.
Kasatkina and Kudermetova, who are both 25, played against each other from a young age and were part of the Russian squad who won last year's Billie Jean King Cup.
- All-Russian clash -
"We played so many times," said Kasatkina. "We were on one team winning Billie Jean King Cup last year, so we have many good memories to share together."
The draw guarantees at least one Russian semi-finalist in Paris.
That heightens the prospect of the newly-crowned French Open champion not being able to play at Wimbledon, where Russian and Belarusian players have been banned over Russia's invasion of Ukraine.
Later Monday, world number one Iga Swiatek faces rising Chinese star Zheng Qinwen while Daniil Medvedev also targets the last-eight.
Swiatek, the Roland Garros champion in 2020, is on a 31-match winning streak and bidding to reach the last eight for a third straight year.
Swiatek, who has won WTA 1,000 titles this season at Indian Wells, Miami and Rome, has not even dropped a set in her last nine matches.
But she may face a tougher test against Zheng, a 19-year-old who has won five second-tier ITF titles since the start of last year and has quickly climbed to 74th in the WTA rankings.
The winner of that match will face American 11th seed Jessica Pegula who downed Romania's Irina-Camelia Begu 4-6, 6-2, 6-3.
Monday's night match on Court Philippe Chatrier sees world number two and US Open champion Medvedev, a quarter-finalist in 2021, play Marin Cilic.
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