Moore made a five-foot birdie putt on par-5 18th hole after Woodland was short with his chip from just off the green.
Woodland and Moore had finished the regulation tournament at 14-under 274 on Sunday with darkness descending on the Kuala Lumpur Golf and Country Club, forcing the play-off to be postponed to Monday morning. The players endured more than three hours of delays on Sunday.
The tournament was an official PGA Tour event for the first time. Moore received a two-year PGA Tour exemption and a spot in the Masters.
The 30-year-old American won for the third time on the PGA Tour. He also won the 2009 Wyndham Championship and 2012 Shriners Hospitals for Children Open.
On Sunday, in a steady rain with shadows being cast by floodlights, Woodland had a chance to win on the 18th hole but missed a 10-foot birdie putt just wide by an inch. Woodland settled for a three-under 69, and Moore had a 70.
Kiradech Aphibarnrat and Chris Stroud finished a stroke back.