Two strokes behind the pacesetting Watson overnight, Stadler birdied his first three holes before recovering from a double-bogey at the 11th on the way to a three-under-par 68 for a 16-under total of 268 at the TPC Scottsdale.
Stadler narrowly missed a birdie attempt from 10 feet at the par-four 18th, then watched nervously as Watson missed a five-footer for par that would have forced a playoff.
Watson, seeking his first PGA Tour victory since his breakthrough major win at the 2012 Masters, had to settle for a closing 71 and a share of second place at 15 under with Canadian Graham DeLaet (65).
Stadler expressed relief and delight after clinching his maiden title on the U.S. circuit on his 239th start, earning a spot at the Masters in April where he will compete along with his father Craig, a former champion at Augusta National.
"It wasn't pretty," Stadler, 33, told Golf Channel after a final round that included five birdies and his double at the par-four 11th. "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."
Stadler and his father Craig, who won the 1982 Masters along with 12 other PGA Tour titles, became the ninth father-son combination to triumph on the U.S. circuit.
Watson held a two-shot lead overnight but was quickly caught at the top by the fast-starting Stadler who sank birdie putts from 10 and 26 feet at the par-four first and second.
Though Watson missed his birdie attempt from eight feet at the second, he and Stadler recorded two-putt birdies at the par-five third to get to 16 under.
Watson immediately regained a one-stroke advantage after hitting a superb tee shot to five feet at the par-three fourth and knocking in the putt.
However, the American left-hander succumbed to a two-shot swing at the par-four ninth where he bogeyed after finding a bunker with his approach and Stadler birdied from three feet to move one stroke clear.
Out in four-under 31, Stadler then tumbled out of the lead with a double at the 11th where his tee shot ended up in the desert with his ball perched on a cactus and he three-putted from long range.
Watson did well to salvage par at the long 15th after driving into water but bogeyed the par-three 16th, where he found a bunker off the tee, to slip back into a tie for the lead with Stadler and DeLaet.
Both Stadler and Watson birdied the par-four 17th after driving the green to edge a stroke clear and it seemed a playoff was likely as they headed for the 18th tee.
Watson appeared to be in trouble after hitting his tee shot into the right rough and overshooting the green with his second but he did well to hit his third to five feet before missing the putt for par.
Stadler, ideally placed in the right fairway off the tee, struck a superb approach to 10 feet and though he failed to sink his birdie attempt, a closing par was good enough to clinch a one-stroke victory.
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