Stroud had chipped in from 51 feet on the 18th hole to force the playoff.
But the 44-year-old Duke made the better approach shot on the second extra hole, bouncing his ball in front of the flag and rolling it close.
Duke wouldn't have been there at all had luck not intervened on the 10th hole, when his ball ricocheted off a tree and onto the green to about 5 feet from the pin, allowing him to make birdie.
Canadian Graham DeLaet finished a stroke back in third place. Bubba Watson finished fourth, two-shots behind, after making a six on the par-3 16th hole.
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