The 30-year-old, son of 1986 US PGA Championship victor Bob Tway, beat Ryan Moore and Brandt Snedeker after all three finished on 14 under. He birdied the 17th and 18th holes in regulation to get into the playoff with Ryan Moore and Brandt Snedeker, then birdied the 18th hole two more times to start the playoff before the players movee to the 10th hole.
Phil Mickelson was unable to get going on the weekend, following up his 74 with an even-par 72 to end up tied for 17th, while Brendan Steele (75) was five shots further back in his bid for a three-peat.
Martin Laird was the highest placed Briton but fell 31 places to joint 46th with a round of 76, including a double bogey on the par-3 second.
Kevin Tway made a 10-foot birdie putt on the third hole of a playoff Sunday to win the season-opening Safeway Open at windy Silverado for his first PGA Tour title.
Snedeker won the Wyndham Championship in August, and for much of the afternoon in Napa he appeared in line for what would have been his 10th career victory.
At 18, his putt from off the green left him a tap-in for his fifth birdie of the day and a 16-under total of 200. "I'm kind of at a loss for words right now". "I played 63 holes of really damn good golf and nine holes I'd like to have back".
Snedeker said he'd remember the event as one he "gave away".
"I was just trying to put the ball in the fairway, which was very hard", Tway said.
The trio finished regulation at 14-under 274.
Aaron Baddeley, who earned a spot in the tournament through the Monday qualifier, birdied the last hole to shoot 69 and finished one stroke back in a tie for fourth with Luke List (67), Troy Merritt (68), Sam Ryder (69) and Sungjae Im (71) of South Korea.