Norrie, who collected his first ATP title of 2023 and fifth overall, looked to be heading for a second straight defeat to the clay court tournament's defending champion when he was broken in the second set.
But with U.S. Open champion Alcaraz needing his right leg wrapped midway through the set Norrie seized the momentum and won four straight games to level the match.
After twice trading breaks in the third set, Alcaraz saved two break points in an epic ninth game but Norrie converted a break point opportunity at the fourth attempt in the penultimate game before claiming the win at the ATP 500 event.
"I was looking done there and I managed to flick a switch and turn it around, so it was a good day," Norrie, who lost to Richard Gasquet in the Auckland final last month, said in remarks provided by the ATP.
"I had to battle a lot of demons the last couple weeks, but I managed to just play well in the big moments and that's what it took today."
Alcaraz, who spent more than three months on the sidelines due to abdominal and hamstring injuries, also needed treatment for an upper left leg issue during his semi-final victory over qualifier Nicolas Jarry on Saturday.
"It was the end of two tough weeks of tennis and of course you're not at your top (level) physically," Alcaraz said.
"I felt something in my leg and that didn't really help me. Now it's time to recover."
Alcaraz said the injury would need to be assessed but did not close the door on competing in Mexico next week.
"I really like Acapulco. I want to go there and play in front of the Mexican fans," he said.
