The Welsh ace had to battle back from behind against both Simon Whitlock and Mensur Suljovic just to reach the Wolverhampton final.
He also trailed by three legs twice against Anderson, at 7-4 and 11-8, before hitting back to lift the Eric Bristow Trophy and claim the £110,000 title.
Price began the decider with a 120 finish, and after finding himself 4-2 down following a three-leg burst from Anderson, the Welshman finished 137 to pull back.
Anderson, though, extended his advantage to 7-4 before seeing his rival claim four successive legs to regain the edge at 8-7.
The world number four found another gear to reel off four legs in a row as he moved back in front, levelling on tops, finishing 72 on the bull, 84 and 144 in taking an 11-8 advantage.
Price finished 70 and 78 to respond before a 130 checkout saw him move within touching distance at 12-11.
Double ten saw Anderson move 13-11 up, but Price again dug in to win four successive legs as he turned the game on its head once again to move a leg away from glory, which was sealed on tops.
Price also picked up a £2,500 group winner bonus as he erased the memory of defeats in the UK Open and World Cup of Darts finals to claim big-stage glory for the first time.
“I’ve got over the winning line and I’m thrilled to bits,” he said. “It was an up and down final but I knew that if I stuck in there I’d get my chances, and I took them.
“I was pretty confident at the beginning of the week and I knew that I had a good chance of reaching the final, but I was chasing Gary all the way there.
“He wasn’t playing brilliantly, he was being consistent at what he does. I knew if I just put it up one gear that I’d beat him.
“I knew that he doesn’t like the way I play. My game is my game, he just needs to concentrate on his own. I’ve won the tournament and I’m over the moon.”
Price added: “I’m the first person to win the Eric Bristow Trophy and that means a huge amount to me – it’s something nobody can ever take from me.”
Price had previously played rugby union and had a brief stint in rugby league before opting to take up darts in 2014 when he won a PDC Tour Card at Qualifying School.
“I’m loving playing on stage against the best players in the world,” he added. “It’s only a few years since I was playing rugby, so to be winning big titles is huge for me.
“Maybe 18 months ago I was scared of the top four or top eight, but now I’m turning up to win these tournaments.”
The defeat saw Anderson miss out on the Grand Slam title for a second time, following his loss in 2011 final to Phil Taylor.
He had earlier ended Van Gerwen’s reign as champion with a sensational 16-12 semi-final triumph, landing five ton-plus finishes to overcome the world number one.
“I played a very poor game,” said Van Gerwen, speaking to Dutch broadcaster RTL7. “I was horribly irritated by myself, and he didn’t play his best game also.
“Gary played ok – he did the right things at the right times and I couldn’t answer that. I missed too many doubles here and there and I can only blame myself. It was just not good enough.”
2018 bwin Grand Slam of Darts
Sunday, November 18
Gerwyn Price 16-12 Mensur Suljovic
Gary Anderson 16-12 Michael van Gerwen
Gerwyn Price 16-13 Gary Anderson