Tway had looked an unlikely victor earlier in the day as Snedeker increased his third-round lead from three shots to five during a solid front nine.
The remaining duo repeated that before heading to the par-four 10th, where Tway made another birdie to clinch the crown.
Phil Mickelson, who started the day sharing second three strokes behind Snedeker, had his struggles, including a double-bogey six at the eighth on the way to a two-over 74.
"I played well on the front nine, but lost it on the back", said Snedeker, who was seeking his 10th PGA Tour victory. "I just did a great job of staying patient knowing the last three holes were birdies holes". After another two birdies on the par-5 18th, Tway ended things with another birdie on the par-4 10th hole.
He had birdied 17 and 18 in regulation to complete a one-under par 71 at Silverado Resort in Napa, California, finishing tied with Moore and Snedeker on 14-under par 274.
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And despite still leading by four shots at the turn, he proceeded to throw away the chance of a 10th career victory as he dropped shots on 10, 11 and 12.
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 have to find a way to win today", Snedeker tweeted. "It sucks right now, but I'm playing next week".
"I was just trying to put the ball in the fairway, which was very hard", Tway said.
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.