Moore (2-under 70) and Woodland (69) each ended 72 holes at 14-under-par 274 Sunday, forcing a Monday playoff after darkness halted play following a pair of lengthy rain delays.
Woodland had a chance to win the event on the 18th hole, but missed a 12-foot birdie putt try while Moore carded a par from in front of the green.
On the playoff hole, Woodland found the rough on his lay-up shot while Moore put his approach within 4 feet of the pin.
Woodland again landed in the rough with his third shot and his chip stopped just a foot shy of the hole, leaving Moore free to drain a birdie for his third career PGA win.
Third-round co-leader Chris Stroud managed a 71 Sunday, ending in a tie for third place at minus-13 with Thailand's Kiradech Aphibarnrat (70).
Charles Howell III (68), Harris English (69) and Graham DeLaet (71) shared seventh place at 10-under.