The 37-year-old Arbroath community policeman held his nerve in a tie-break, winning it 2-1, after the pair could not be separated following two closely fought sets.
An elated Burnett said: "I am totally drained. It was a topsy-turvy game and neither of us played the way we can. I was really nervous, which I never usually am, but it was my first world indoor title at this level so you've got to expect something.
"I scratched about a bit in the early stages and I couldn't quite master the weight. I think we both experienced that. As the match progressed I gradually lost that tension and felt quite relaxed."
King, an estate manager from Norfolk, got on top in the early stages of the opening set with three singles, before Burnett hit back with two singles and a double to go 4-3 ahead.
However, King continued with his trademark drawing prowess and rolled in four successive singles to take a 7-4 lead. Requiring three shots to square the set or four to win, Burnett held two and the possibility was there for him to trail the jack slightly to take that magical count. But he opted to play a runner to take the jack to the ditch and paid the price when he only counted a single, losing the set 7-5.
In the second set, Burnett was staring down the barrel at two shots down after two ends and a count of three against him, but he played a stunning running shot to clear King's bowls out of the head and in turn claim a double to get him on the card. Another turning point came in the ninth end when he counted a treble to go 8-6 ahead and although King levelled with a double, Burnett grabbed the vital lifeline on the last to win 10-8.
In the first end of the tie-break and with a shot against him, Burnett played a runner and won the measure with an inch to spare to a displaced jack. King came back to take the second but in the third and final end, Burnett beat King's first shot by a few inches and his bowl stayed in place until the finish to earn him the title.
Burnett added: "I know I played aggressively at times but I always promised myself that if I ever reached this final then I would do what I had to do to win the title. Over the years I've stuttered about at Potters so I'm just so pleased to have that monkey off my back. You know you've got the game but you never know if you will win it. I knew that I was going to take every opportunity that came my way and not have any regrets.
"Mervyn is the local boy so I was conscious that I would have to make an impact in the early stages to try to silence the crowd. But it never really panned out like that. The atmosphere was electric and despite the long distance from Scotland to Norfolk, I had a magnificent support too.
"It's been a long haul getting to this title but I was pleased to share it with my wife Linsey and daughters Isla and Evie. They flew down midweek and it was good to get some family time with them and they helped me to relax.
"Over the piece, though, I've felt that I just gained a bit more confidence as each game progressed. I probably played my best game against Rob Paxton in the semi-finals but in his semi-final Mervyn just played with relentless precision against Paul Foster, so I was under no illusion just how well I would have to play to beat him."
King, though crestfallen to have lost in front of his home crowd, admitted his opponent had played better at important times. He said: "I was just so disappointed with my last delivery to save the title. Over the piece Darren played some really big bowls. I felt I was more consistent but that doesn't mean much at the end of the day."