Watson, who birdied four in a row on the back nine, closed with a 68 to join Oosthuizen at minus-10. Oosthuizen carded a 3-under 69 in the final round to get into the extra session. Four players were two strokes back, including third-round leader Peter Hanson, who shot 1-over 73 to slide down the leaderboard.
Oosthuizen, the 2010 British Open champion, posted the fourth double-eagle in Masters history on the par-5 second. However, a poor drive on the second playoff hole cost him.
He came up short with his second shot, then chipped his third to 12 feet. Oosthuizen's par putt slid by the hole and opened the door for Watson, who needed to 2-putt from 10 feet out.
And that is exactly what Watson did. He rolled his birdie try inside a foot and tapped in for the win.
The devout Christian let the tears flow as his first major championship victory came on Easter Sunday.
Among the four players two strokes out of the playoff was Phil Mickelson, who stumbled to a triple-bogey on the fourth. Mickelson played not one, but two shots right-handed on that hole. He needed an eagle at the par-4 18th to get into the playoff, but a poor drive cost him.
Hanson, who made the cut for the first time at The Masters, also shared third and that was his best finish in a major.
Four-time Masters champion Tiger Woods had his worst-ever finish as a pro at Augusta last year. He failed to break par in any of the four rounds, and ended in a tie for 40th at 5-over par.
Woods has won three of his five PGA Tour starts this year, and six of his last 20 overall, so he comes in playing well.
Mickelson, a three-time Masters champion, has had an inconsistent year, but did win in Phoenix and shared third at Doral.
ESPN has coverage the first two days, before they hand off to CBS for the final two rounds.
The PGA Tour heads to South Carolina next week for the Heritage, where Carl Pettersson had a 5-stroke win last year. The European Tour heads to Spain for the Open de Espana, where Francesco Molinari fired a 65 in the final round to win by three over a trio of players.
WNB GOLF CLASSIC, Midland, Texas - The Web.com Tour heads to Texas this week for the WNB Golf Classic, where Luke Guthrie rallied for a 1-stroke win over Cameron Percy and third-round leader Danny Lee.
Lee was also the 2011 champion, and entered the final round looking to become the first player in tour history to win the same event in consecutive years. Lee had a birdie and a bogey on the front nine.
On the back nine, he carded three birdies and three bogeys in the first six holes. Lee parred the final three to end one back. He became the fourth player in tour history to finish second in a title defense.
Guthrie birdied the first three holes, then had a bogey and four birdies the rest of the way. That was good enough for the win when Percy bogeyed 16 and 17 to fall into a share of second place.
Lee is back in the field this week, as are the top-10 players from the tour's money list.
There is no television coverage this week.
After a week off, the tour heads to Georgia for the South Georgia Classic, where Luke List won by two strokes last year.