Jarry, who fought off 11 of 13 break points faced in the match, will next play American 14th seed Tommy Paul on Friday in his first ATP Masters 1000 semi-final.
Another Chilean, Alejandro Tabilo will take on German third-seed Alexander Zverev in the other semi-final. Two Chileans will appear in the semi-finals of an ATP 1000 event for the first time since the format was introduced in 1990.
Jarry was tested in the opening set by Greece's sixth seed but gained his momentum in the second.
The 28-year-old won five consecutive points at 3-3 as he bounced back from 0-40, before wrapping up the second set by winning nine of the final 11 points.
Jarry then broke Tsitsipas in the decider for a 5-4 lead to eventually cruise into the last four.
"Beating Stef on clay is a good achievement. Extremely happy for my fight today and how I played. How I maintained myself even down a set, down a break," Jarry said.
"I just kept trying to find different ways to have chances on his serve. I knew I was playing good.
"I had to adjust some things on the backhand so he didn't push me back. That's why he played so good. I was able to do those two things... It was a good fight tonight. Hopefully tomorrow I can be stronger."
Earlier on Thursday, Paul came through a marathon match to beat Polish seventh seed Hubert Hurkacz 7-5 3-6 6-3 to book his spot in the last four.
The American, who defeated defending champion Daniil Medvedev in the previous round, had yet to drop a set before the quarter-final, but Hurkacz managed to force the match into the decider before Paul came out on top.
It was a match which went back and forth with 13 breaks of serve overall. The pair exchanged breaks early in the opening set, with Paul breaking twice to lead 3-2. With Paul serving for the set, Hurkacz broke again to make it 5-5.
Paul broke back immediately and held serve to take the first set. The American made a strong start to the second, breaking in the first game, but Hurkacz came back, breaking twice to force a third set.
This time it was the Pole who took an early 2-0 lead, but they again exchanged breaks of serve to make it 3-3 before Paul made the decisive break to go 5-3 up, and then saved six break points to wrap up victory.
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