A commanding three up lead after 12 holes on an afternoon of light breezes at Dove Mountain, Australian Day was caught by the Frenchman, who recorded clutch birdies at the 13th and 17th, followed by an amazing par from a greenside bunker at the 18th.
Dubuisson then conjured two miraculous up-and-downs from desert scrub at the 19th and 20th holes to keep the match alive before Day sealed the win by sinking a four-footer for birdie at the 23rd, the driveable par-four 15th.
The Frenchman, who had missed the green to the right off the tee, hit a high lob wedge 20 feet past the cup and missed the birdie putt coming back before the Australian made his to secure the title.
"I've never wanted something so bad in my life," an emotional Day, 26, told CBS Sports after clinching the winner's cheque for $1.53 million.
"Obviously I didn't want it to go this long but Victor, he has a lot of guts and he has got a great short game. He got it straight out of a cactus twice.
"The biggest thing was just how much do I want it, how much do I want to win? I kept saying that to myself last night, kept visualising myself with the trophy. I am just glad that I could finish it off, but it was a close one."
It was Day's second triumph on the PGA Tour, following the 2010 Byron Nelson Championship, and his first success in one of the elite World Golf Championships (WGC) events which bring together the game's leading players.
Dubuisson, who had never previously played matchplay golf until his WGC debut this week, appeared to be on the way to a heavy defeat after he recorded five bogeys in the first seven holes to reach the turn three down.
However, the 23-year-old Frenchman raised his game on the back nine and trimmed the deficit to two with a birdie at the par-five 13th where he was just off the edge of the green in two shots.
Both players parred the short 16th before Dubuisson won the par-four 17th with a magical birdie to trail by just one.
The Frenchman, who booked his place in the final with a one-up victory over veteran South African Ernie Els earlier on Sunday, had ended up in a fairway bunker off the 17th tee, but struck a stunning approach to 12 feet above the hole.
Day, in the right rough off the tee, hit his second shot to 20 feet and narrowly missed his birdie attempt before watching steely-eyed as Dubuisson coolly sank his putt for a three.
The Frenchman's late bid for the title looked decidely shaky at the par-four 18th when he pulled his approach well left into a greenside bunker.
Day, who beat American Rickie Fowler 3&2 in his Sunday semi-final, was on the back fringe of the two-tiered green in two, 70 feet above the hole, from where he ran a very slick first putt 10 feet past the cup.
Dubuisson conjured a miraculous shot from the sand for his ball to end up five feet away and the Australian, with the title on the line, left his par putt inches short of the cup.
Ice-cool, the Frenchman knocked in his par putt to level the match and force extra holes.
Two astonishing moments were produced by Dubuisson before Day finally clinched the title.
At the 19th, the par-four first, the Frenchman ended up in desert scrub over the back right of the green with his approach.
Day then struck a wedge into the left greenside bunker before Dubuisson, using soft hands, somehow hit his third shot from behind a cactus, through sand, rocks and a television cable strung in front of him, to four feet.
After the Australian had splashed out to six feet, he sank the par putt before Dubuisson followed suit to remain all square.
One hole later, at the par-four ninth, the Frenchman missed the green badly to the left with his approach, his ball ending up in a bush from where he amazingly hit his third to seven feet, prompting a wry smile from Day.
The Australian was on the back fringe in two from where he cosied his approach putt to within two feet of the cup. Dubuisson, non-plussed, duly sank his par putt to keep the match alive.
"I know he was the (world's) number one amateur back in 2009," said Day about the gutsy Frenchman. "He has got a lot of game and you're going to see a lot of him for years to come."
- Sports & Recreation