The American clinched his first major title with a two-putt par on the second extra hole, the par-four 10th, where South African Oosthuizen bogeyed after ending up just short of the green in two and failing to get up and down.
Both players had ended up well right off the tee, Oosthuizen gaining a fortuitous bounce off a tree before playing his second shot from the first cut of rough.
Although the long-hitting Watson ended up deep in the tree line, he had an avenue to the green and struck a superb high draw off the pine straw with a gap wedge for his ball to settle 10 feet from the pin.
After Oosthuizen had chipped up 15 feet past and narrowly missed his par putt coming back, Watson had the luxury of two putts for victory.
"I never got this far in my dreams," a teary-eyed Watson said in the Butler Cabin before being helped into the revered green jacket by last year's Masters champion Charl Schwartzel of South Africa.
"I got in these trees, hit a crazy shot that I saw in my head and somehow I'm here talking to you with a green jacket on.
Asked to explain how he had pulled off his stunning approach on the second playoff hole, Watson replied: "I had a good lie, had a gap where I had to hook it 40 yards or something.
"I'm pretty good at hooking it, so I just hooked it up there and somehow it nestled close to the hole."
A self-taught golfer who learned the game by hitting wiffle balls around his house, Watson became the eighth consecutive first-time winner of a major and the 11th in the last 12 championships played.
Oosthuizen, the 2010 British Open champion who had earlier spectacularly grabbed a two-shot lead with a stunning albatross two at the par-five second, paid tribute to Watson.
"We had a great day," Oosthuizen told reporters. "It's fine, he had an unbelievable shot there. I played well ... but great stuff from him and he deserves it."
The duo had finished the regulation 72 holes on 10-under-par 278, Oosthuizen carding a three-under 69 and Watson drawing level with a sizzling run of four birdies from the 13th on the way to a 68.
British world number three Lee Westwood birdied four of the last six holes for a 68 to share third place at eight under with Americans Mickelson (72) and Matt Kuchar (69), and Swede Peter Hanson (73), the overnight leader.
Four-times winner Woods made a humbling exit from the tournament after battling his way to a 74 and a five-over total of 293, his worst Masters finish as a professional.
On a glorious sun-splashed afternoon, cheers repeatedly echoed around the Georgian pines as the fans were treated to dazzling shot-making and a few unexpected blunders with five different players holding at least a share of the lead.
While American Bo Van Pelt and Australian Adam Scott each recorded a hole-in-one, Oosthuizen sparked some of the loudest roars ever heard at Augusta with his remarkable albatross at the second, where he holed out from 253 yards with a four-iron.
The gap-toothed South African watched as his ball pitched just short of the green before bouncing and then rolling some 50 yards and curving left to right before dropping into the cup.
Oosthuizen thrust both arms skywards before high-fiving his caddie, having recorded the first albatross, or double-eagle, at the second hole and only the fourth ever at the Masters.
Although Oosthuizen bogeyed the fourth and the 10th for his lead to be cut to one, he regained control with birdies at the 13th and 15th and several clutch par putts before being caught by the charging Watson.
Fan favourite Mickelson had been one stroke behind overnight but he left himself with too much ground to make up after an adventurous triple-bogey at the par-three fourth.
After his tee shot deflected off the grandstand into bamboo trees left of the green, the American hit two poor right-handed shots then struck his fourth into a bunker from where he got up and down for a six.
Woods rued his tendency to allow old habits to creep back into a swing he has grooved with coach Sean Foley over the last two years.
"This is a golf course you just have to dominate the par-fives, and I did not do that at all this week," the 14-times major champion said after mixing five bogeys with three birdies.
"I fall back into the same old patterns again and I just need to do more reps. Thank God my short game was good this week and my putting was really good. Unfortunately they were all for pars, not for birdies." (Reporting by Mark Lamport-Stokes; Editing by Nick Mulvenney)