Although Day never trailed, he couldn't shake Dubuisson, who first rallied from a 3-down deficit to force extra holes and then came through with two extraordinary par saves to extend the contest.
Dubuisson made a save from under a bush on the 19th hole and came through again with a tremendous approach shot from the desert on the 20th at the Golf Club at Dove Mountain.
Day finally wrapped it up with a routine up-and-down birdie on the 23rd after the Frenchman had struck a chip shot that rolled 28 feet past the hole.
The victory was Day's second in PGA play and first in a World Golf Championship event.
Notable deaths of 2014 [PHOTOS]