The Spanish teenager collapsed on his back after the more than five-hour match concluded at 2:50 a.m, the latest finish in US Open history.
The win keeps Alcaraz's hopes alive of winning a maiden Grand Slam title and claiming the world number one ranking.
"I don't understand how I did it," he said in an on-court interview. "The level that I played, the level of the tennis, it's incredible."
Alcaraz trailed by two sets to one after taking the first set, and Sinner had a golden opportunity when serving for the match at 5-4 in the fourth. However, after squandering a match point, Alcaraz hit back to claim the fourth set.
The American crowd were clearly on the side of the Spaniard, but the Italian gained the initiative once more with an early break in the final set.
Alcaraz put his finger to his ear in the fifth set after he raced across the court to catch up with a Sinner drop shot and fired it past the Italian to set up break point, bringing fans at Arthur Ashe Stadium to their feet.
Alcaraz converted on the next point when Sinner's shot hit the net for a 5-3 lead and then fired a serve into Sinner's body on match point to end the five-hour, 15 minutes contest.
"Thank you to all of you," he said to the crowd.
Alcaraz was a human highlight reel all night, sliding and diving around the court and hitting a sensational behind-the-back circus shot that set up a forehand winner in the second set.
If the match was a preview of the future of men's tennis, the sport is in good hands with the 21-year-old Sinner and 19-year-old Alcaraz engaging in one thrilling exchange after another to the delight of the fans.
Sinner struggled to sum up the disappointment.
"It's not easy to talk now because I finished the match," he told reporters.
"It was a good match from my side, from his side.... the level was high.
"But very, very tough."
Alcaraz will take on home favourite Francis Tiafoe for a place in the US Open final, while Karen Khachanov and Casper Ruud will battle it out on the other side of the draw.
