The German's victory earned him a meeting with defending champion and top seed Carlos Alcaraz.
Zverev will look to keep his superb run going having hit top gear again after a serious ankle injury cut his 2022 season short.
"I guess I can say I'm back. This is what I live for, this is what I absolutely love to do," Zverev said.
"I would have loved to play a little shorter, that's for sure, but last year when I wasn't able to play, this is what I missed. Playing past 1:30 a.m. in front of a packed crowd on Arthur Ashe Stadium. There's nothing better."
Zverev swapped breaks with Sinner early in the contest and was locked in a tight battle until 4-4 before going ahead to take the first set with two aces.
Sinner responded immediately with some heavy hitting in the next set to go up 3-0 and turned up the heat in a steamy Arthur Ashe Stadium to thwart the 2020 finalist's late charge.
Sinner then struggled with cramp in both legs in the next set. The Italian saved five breakpoints to hold for 2-2 but 12th seed Zverev reeled off the next four games.
The World number six began swinging at everything in the fourth set and Zverev cracked under the onslaught, dropping serve in the ninth game as his opponent forced a decider.
But a crucial break in the fifth set gave Zverev the upper hand and he held his nerve to close out the win in four hours and 41 minutes with a powerful serve.
Zverev said he was "completely done" after the fourth set.
"I was extremely tired. He was fitter than me even though he was cramping, he was in better shape. In the fifth set I somehow found it again," he added.
"Without the energy, without the crowd this wouldn't have been possible. This is one of the best moments of my career after my comeback, after everything. I'm looking forward to what's next."
A daunting challenge awaits in the last eight but Zverev said he would be ready.
"I'll give it my absolute best like I always do," Zverev added. "I'll fight until the last moment. Of course, I have to recover after this physical match but I'll be ready."
