The world number one had to "dig deep" to contend with an in-form Ernie Els and shot two birdies at the last two holes to take the tournament to a play-off.
Woods had not been playing well off the tee all day, but mid-round tweaks helped him recover some of his old poise in a nail-biting final few holes.
"I was very fortunate today," he said. "Starting out it was not good because I had a two-way miss going, meaning I could hit it left or right at any given time, and that's not a whole lot of fun.
"Somehow I just hung in with my short game and then I fixed it on the back nine, where I hit some really good shots.
"I was digging deep before the last two holes. To be honest, anyone can win when they are playing well.
"It's when you have an opportunity to fix things, which I did, turn it around and somehow end up on top. That's always satisfying."
Woods' eye for improvisation was never more evident than at the 14th hole where he found a rocky lie off the tee yet still managed to make his escape, eventually holing from 12 feet for par.
"It was important to make that par," the American stated. "Ernie at that time was at 17 under or something and I was one back.
"If I'd fallen two back with 17 and 18 pretty easy birdie holes to come I would have had to add another birdie in there somewhere, either on 15 or 16, and that was not an easy task.
"The par allowed me to stay only one back where anything could happen on 17 and 18 and I could still get into a play-off and maybe win it outright."
Els was philosophical in defeat, owing to the perspective of possibly never playing golf to his former standard. The world number five wrote off much of 2005 with a knee injury that could have ended his career and, with that in mind, Els was satisfied with second, even if it meant missing out on a fourth Desert Classic crown.
His superb six under par 67 was not enough to win outright after Woods' late heroics, but he blew his shot at winning with a wayward drive off the tee at the replayed 18th and followed it up by finding the water hazard.
"After all the hassle I had with the leg, to come back and almost win, with the strongest field in the world this year, is fine," he said.
"I didn't do what I was supposed to do in the play-off. That was my favourite tee shot of the whole tournament. I was five under on that hole for the week, an eagle and three birdies, but there you go. I thought a 65 would have won it beforehand. I missed a five-foot putt on 13 for birdie, and a few other ones, but that was the biggest."
There was heartbreak for Richard Green, whose victory at the Desert Classic in 1997 marked the last time the Australian tasted success on Tour, and he will get few opportunities as good to end the nine-year drought.
Green was one clear at 19 under going to the last after making birdies at 13, 15, 16 and 17, but took a six at the 18th, leaving him to lament the problems faced being one of the few southpaws around.
"Unfortunately, my driver has cost me another win," he rued.