Stadler ended at 16-under-par 268 to corral his first title in his 239th career start.
The 33-year-old son of former Masters champion Craig Stadler avoided a playoff when Watson was wide with a 5-foot putt for par on the final hole to wrap up a even-par round of 71 at TPC Scottsdale.
Graham DeLaet (6-under 65) shared second place at minus-17.
Knotted for fourth at 14-under were Hunter Mahan (68) and Hideki Matsuyama (69).
Senate Democrats to pull all-nighter on climate change
Susan Sarandon 'very excited' about daughter's pregnancy