Second seed Norrie came from behind to claim the fifth tournament win of his career, winning 5-7, 6-4, 7-5 in 2hrs 40mins.
Norrie's victory came exactly seven days after the British world number 13 had lost to Spain's second-ranked Alcaraz in the final of the Argentina Open in Buenos Aires.
"It's so special to win this one, especially after losing a couple of finals this year," said Norrie, who was also beaten in the final in Auckland in January.
But while Norrie celebrated, Alcaraz was left with fresh fitness concerns after an energy-sapping duel that saw him require repeated visits from the physio.
The 19-year-old, who only returned to tennis at last week's tournament in Argentina after a four-month injury layoff, was clearly struggling with a sore hamstring during the latter stages of Sunday's final.
He needed heavy strapping on his right thigh midway through the second set, when he collapsed from a 3-0 lead to allow Norrie to take the set and level the match.
Alcaraz is due to play at the Mexican Open in Acapulco next week and after that has the Indian Wells Masters looming in California in early March.
The former world number one had looked to be in control after clinching the first set with a rasping forehand up the line to break Norrie for the first time in the match.
He then sprinted into a 3-0 lead with another quick break of serve in the second set and appeared to be building unstoppable momentum.
However, Alcaraz was soon struggling with his sore thigh and Norrie broke twice to take a 4-3 lead.
Although Alcaraz fought back to break again and level at 4-4, the Spaniard was broken on his next service game and Norrie held to level the match at a set apiece.
While Alcaraz was clearly not firing on all cylinders in the decisive third set, the Spaniard was still capable of moments of free-swinging inspiration with 19 forehand winners.
But Norrie gave himself the chance to win his first tournament of 2023 after breaking Alcaraz for the sixth time in the match for a 6-5 lead.
Norrie then held, clinching victory with an ace.
"I had to do it the tough way -- a set down and a break down and 0-30 on my serve," a jubilant Norrie said afterwards.
"I was looking done there, but I managed to flick a switch and turn it around so it was a good day.
"I had to battle a lot of demons, but I managed to play well in the big moments and that's what it took today."
© 2023 AFP