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."
Auger-Aliassime advances to a fourth-round meeting with Russian 15th seed Andrey Rublev on Monday.