Tsitsipas, the fifth seed, won 6-3, 6-3, 4-6, 4-6, 6-3 after falling in the semi-finals of the Grand Slam tournament in each of the previous three years.
In the last set, Tsitsipas found his form again and was able to hold off Zverev, but he lost four match points before finally serving out.
On Sunday, the 22-year-old will play either Novak Djokovic or Rafael Nadal for the championship.
Top-seeded Serb Djokovic faces defending champion and potential record-breaker Rafael Nadal of Spain later on Friday.
At the net, Tsitsipas and his opponent embraced after Tsitsipas served an ace to win the match on his fifth match point.
The Greek sat back in his chair, taking in his accomplishment, before he became visibly moved during the first few minutes of his on-court interview.
Tsitsipas, after gathering his thoughts, said: “My mind keeps returning to my beginnings and family tree. I grew up in a tiny town not far from Athens.
“My ultimate goal has always been to compete at the French Open. To think I actually did it is beyond my wildest imagination.”
Tsitsipas triumphs despite a wobbly start and a “intense” last set.
As two of the men’s game’s brightest players on the rise, Tsitsipas and Zverev are poised to become big rivals when the ‘Big Three’ of Djokovic, Nadal, and Roger Federer retire.
In their young careers, both men have already won the season-ending ATP Finals and Masters titles, but they have yet to win one of the four major titles.
The Match was Decided in the Opening Game of the Deciding Third Set.
After Zverev had gained some momentum, Tsitsipas saved three break points and went on to capture the game-winning break in the fourth.
Tsitsipas, who won in three hours and 37 minutes, said, “The opening game was nerve-wracking and very dramatic – it was the most critical game in the fifth set.”
“I made it back, didn’t die, and rallied a large following. Motivated by their presence, I pushed forward.
“There was still time for me to make a change and fight back, I thought. I actually did that!”
Zverev’s Comeback in the US Open was Unsuccessful.
Zverev, unlike Tsitsipas, has already won a Grand Slam semi-final and lost in the final against Dominic Thiem of Austria at the 2018 US Open.
The German battled back from a two-set deficit in his quarterfinal match against Pablo Carreno Busta to reach his first major final in New York and threatened to do the same thing at Roland Garros.
It seemed more likely after he won the third and fourth sets with the help of early breaks, both of which he backed up with excellent serve.
Zverev had played an impressive 23 five-set matches by such a young age in his career. He proved his mental toughness to handle the pressure of another match by winning 16 of those deciding matches.