The 34-year-old reached his first Grand Slam quarter-final since losing the 2017 Roland Garros final to Rafael Nadal, winning 7-6 (8/6), 5-7, 6-4, 3-6, 8-6 after five hours and nine minutes in the longest match of this year's tournament.
The 2015 champion will face close friend Roger Federer next as he looks to improve his 23-3 losing record against his fellow Swiss.
It completed a remarkable turnaround for Wawrinka, who slipped out of the world's top 250 following his first-round exit to Guillermo Garcia-Lopez 12 months ago, after months of struggles with knee injuries.
"I just felt relieved. I enjoyed the atmosphere on the court," he said.
"You know, such a match, so long, with so many fans... and these are emotions that you can only have in our sport, in tennis. And I'm happy I can still feel that."
Greek sixth seed Tsitsipas was left in tears after seeing eight breaks points come and go in the dramatic final set, which ended with Wawrinka's magnificent winner on his second match point from yards behind the baseline.
"I was pretty sure it was in. When I hit it, I saw it was on the line. I was just hoping that the umpire would confirm it," added the three-time Grand Slam champion.
"It was (an) amazing battle today. Today we saw only one centimetre can change the winner. I think he would have deserved as much as me to win the match today.
"It's for sure a tough loss for him. I'm happy with everything that I've done today, the way I was fighting, the way I was staying with him, even if I was tired."
Wawrinka, seeded 24th in Paris this year after steadily climbing back up the rankings, had not even played in the second week of a Grand Slam event since the loss to Nadal two years ago -- after which he took time out from the sport to have surgery.
He dug deep on a raucous Court Suzanne Lenglen, saving 22 of 27 break points in total and hammering 62 winners.
"For sure when I'm on the court I try to enjoy and remember also everything I have done to be here.
"I think I (have) never experienced that kind of atmosphere here in Paris on the Lenglen. Always had a lot of support, but I think today was really special."