Two shots behind the pacesetting Woods at the start of the day, Johnson fell four behind with just eight holes left before making four birdies, along with a spectacular wedge hole-out for par at the last, to take the tournament into a playoff.
Having twice finished second to tournament host Woods in the elite limited-field event at Sherwood Country Club, Johnson reversed that position as he safely parred the first extra hole before Woods surprisingly lipped out from five feet to bogey.
"You want to end the tournament with someone making a putt," Johnson said after Woods' miss handed him his second tournament victory of the year. "You don't want to see it like that, especially when he has hit a really good sand shot."
Woods had dumped his approach into the front right greenside bunker at the par-four 18th before failing to get up and down to extend the playoff.
"He played great," Johnson said of the American world number one, who had previously won the World Challenge five times. "He didn't make as many putts as I did. That's all it really was. Today."
Johnson, whose only major victory came at the 2007 Masters, is a medium-length hitter renowned for his brilliant short game and never-say-die attitude.
"I love the competition, I love being in difficult situations and having to execute and that sort of thing," said the 37-year-old American, a 10-times winner on the PGA Tour.
"I like when you get somewhat in contention, I like getting after it. I've always liked that. Doesn't matter what sport it is.
"I know what my talent is, and I know my limitations and I feel like I know my game. So if it's good enough that week, that day, then great. If it's not, I'll just keep working."
What made Sunday's charge down the stretch even more special for Johnson was that he was duelling with 14-time major champion Woods, the greatest player of his generation.
"Everybody talks about going head to head with him," said Johnson. "That's what I want as a competitor. I want to play against the best. He's the best I've ever played with, and I want to put myself in that position.
"I've been in it, I don't know how many times, twice here and a handful of times on Tour. I like playing with him. He's a friend and he seems to bring out the worst and the best in you, you know.
"If I win, I win. If I lose, I lose. That's kind of the way I approached today. I just like to compete and being in those situations where you get into holes you have to execute, you have to hit a shot. I feel very honoured to be the champion."