Watson was in charge for much of the tournament could only muster a level-par 71 on Sunday to finish 15 under overall.
He was set for a two-man play-off, but bogeyed the 18th to hand the title to Stadler on 16 under.
In claiming his first PGA Tour title Stadler emulated his father Craig, who had 13 wins on tour, including the 1982 Masters.
Winning in Phoenix earned Kevin the right to play in Augusta this year.
Stadler told the Golf Channel: "It wasn't pretty. I was walking all over the shop. I hit a lot of good shots and a lot of kind of ugly shots but thankfully it all worked out.
"It feels fantastic. Obviously I would have liked to make that [birdie] putt there [on 18] and win that way but it's been a long time coming, so I'll take it any way I can get it."
The title looked a long way off when he hit a double bogey at the par-four 11th.
By that time, though, he already had four birdies to his name and he added another at the 17th for a round of 68, which proved to be enough.
Through 15 holes Watson had two birdies and a bogey to his name but he finished bogey-birdie-bogey to surrender the lead to Stadler, throwing away even the chance of a play-off when he missed a five-foot par putt on the last.
In the end he finished in a tie for second with Canada's Graham DeLaet, who carded a 65.
Hunter Mahan and Hideki Matsuyama were tied for fourth on 14 under.
Martin Laird finished on seven under to be the leading European, while 2013 champion Phil Mickelson finished at three under.