Murray had the match on his racket, forcing a number of break points at key times, but he could not convert them and Norrie stole a 3-6, 6-3, 6-4 win.
It was the second meeting between the pair, with Murray winning the first in 2019, but Norrie, seeded ninth, got his revenge and advanced to the last 16.
The Scot left the court having had treatment for thigh cramp but the 11 missed break points will have hurt more.
“I think there was not much in it,” Norrie said on Amazon Prime.
“It came down to a little bit of physicality and it didn’t really help him. He was struggling a little bit towards the end and I just tried to get the ball in at the end.
“I am still struggling with the conditions a litte bit and trying to find rhythm. I just got through with my attitude and stay positive.
“I managed to find myself at one set all, I don’t know how and the last set was a battle, he had some chances and I played a couple of good points.
“That’s how it rolls sometimes in tennis. It’s a massive win for me beating Andy for the first time. It’s not the way I wanted to beat him with him cramping, but it’s nice to get through.”
After there was an exchange of breaks in the first two games, Murray assumed control and dictated the points.
He missed the chance to break in Norrie’s second service game but continued to play on his terms and eventually broke at 4-3 before serving out the next game to take the first set.
The Scot, who was reduced to tears after his emotional first-round win over Stan Wawrinka, had the perfect opportunity to grasp an even firmer control of the match in the opening game of the season as he forced break point.
But with the court gaping he put a routine volley wide and Norrie survived.
And that miss proved costly as Norrie began to knock on the door. Murray was able to save break points at 2-3, but could not repeat the trick at 3-4 as the British No 1 struck to go 5-3 up.
He had 40-0 as he tried to force an immediate break back but Norrie got himself out of trouble and served out the second set.
Murray regrouped and his big moment came in a lengthy sixth game, but he squandered three break points and Norrie survived.
Eventually all those missed opportunities came back to haunt the 35-year-old as a double fault handed his compatriot the crucial break and he served it out to claim a victory that will have tasted very sweet.