Zverev, the 2020 US Open runner-up, claimed the first set and had a set point in the second but came unstuck as Auger-Aliassime won 4-6, 7-6(7), 6-4, 6-4 in three hours and 48 minutes.
It is the first time Zverev has failed to reach the second week in New York since 2018. He suffered his earliest Grand Slam exit in six years at Wimbledon last month after losing in round one.
Auger-Aliassime's best run at a major came when he reached the semi-finals at Flushing Meadows four years ago.
"This feels good. I've been coming here since 2018. I'm still young, it's been a few years, but I'm working my way," said the 25-year-old Auger-Aliassime, who beat a top-five player at a Grand Slam for the first time.
"The tournament's still going, the job's not done, but this means a lot to me."
Zverev edged a tight opener, the German third seed striking early with a break in the opening game.
Though the Canadian clawed it back Zverev immediately pushed again to reclaim the lead and then slammed the door shut to win the set, rattling off three routine holds without facing a break.
The second set turned into a serving showcase with neither player facing a break point as it went to a tiebreak.
Zverev, still searching for his first Grand Slam title, looked poised to pull away after a double fault gave him the edge at 4-4, but Auger-Aliassime again fought back with fearless shotmaking.
Both let set points slip before the Canadian finally snatched it 9-7, a fortuitous net cord sealing the set and squaring the match.
The 25th seed raised his game in the third, breaking early and hitting sharp winners that rattled the German, who slammed his racket as he grew increasingly frustrated as the set steadily slipped away.
Auger-Aliassime took full control in the fourth set as he went on to break to love, seizing a decisive advantage and closing out the set on serve to secure the win in front of a raucous crowd at the Louis Armstrong stadium.
Auger-Aliassime advances to a fourth-round meeting with Russian 15th seed Andrey Rublev on Monday.