The burly American reeled off seven birdies in a seven-under-par 64 to move one stroke clear of Germany's Marcel Siem.
Stephen Gallacher was in third place on 66, one ahead of fellow Briton Jamie Donaldson and Chilean Felipe Aguilar.
"To not make a bogey out there is something I didn't think was possible when I first played it," the 35-year-old Stadler told reporters. "It's instantly one of my favourite courses ever, I thought it was spectacular from hole one to the end.
"They talk about the last four holes but they are doing a discredit to the previous 14. I have always wanted to come over here, the guys in the U.S. tell me it's one of the best events of the year, and it has not let me down - it's been fantastic."
Gallacher, who began his round at the 10th hole, missed a chance to take the lead after dropping strokes at the seventh and ninth.
"I didn't do too much wrong on the last couple of holes but that's what this course can do," said the Scot, who is just outside the automatic qualifying places for Europe's Ryder Cup team.
French Open champion Graeme McDowell opened with a 70 while U.S. Open winner Martin Kaymer started out with a 72.
(Writing by Tony Jimenez, editing by Stephen Wood)
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