Third-seeded De Minaur was up 5-2 and 40-30 in the second set, but stumbled to allow Norrie back into their third-round match.
He had another chance when serving for the match at 5-4, but again double-faulted to surrender the game on Norrie's fourth break point.
World No.37 Norrie then dug deep to record a 5-7 7-6 (7-5) 6-1 victory after two hours 52 minutes in difficult, humid conditions.
Norrie said he had to weather de Minaur's early onslaught with the newly-married Aussie starting with a bang.
"From the beginning, Alex came out firing and I was like, 'Woah, this is a crazy level', and I had to match it," Norrie said in his on-court interview.
"I managed to go up a break (in the first set), but he stayed really tough and I know he's so tough as a competitor and stole that first set.
"He was looking good in the second set, and I think he took the foot off the gas very slightly and I managed to free up, steal the tiebreak and then it was just a battle.
"It was a great match to win, and to test my level and I can take so much confidence from this one."
Entering Thursday's (Friday AEST) match, de Minaur was the highest seed remaining at the hard court tournament, with defending champion and fifth seed Ben Shelton taking over that title.
In some good news for de Minaur, because American Taylor Fritz and Italian Flavio Cobolli have also exited the event early, he can only possibly drop from seven to eight in the rankings - and that's only if Shelton wins the title.
That means the Australian, who broke through for a career-high No.5 ranking in July, will keep his top-eight US Open seeding intact for now.
De Minaur is looking to improve on quarter-final finishes at at Flushing Medows in the past two years.
Norrie will next take on 18th seed Arthur Fils, who earlier cruised past Mariano Navone 6-3 6-2 to maintain his impressive record at Masters 1000 events this year.
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