Snedeker had squandered a five-shot lead during the final round before being eliminated at the first extra hole at the Silverado Resort, Tway then sinking a 10-foot birdie at the third extra hole to beat Moore.
It was the first PGA Tour victory for the 30-year-old, who was born less than two years after his father won the 1986 PGA Championship.
Kevin and Bob are the 10th father-son duo to record PGA Tour victories, Tour data showed.
"I think he's probably crying to be honest," said Tway of his father, who had been watching on television.
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"He's pretty proud of me. I talk with dad every night. He's probably the biggest person I lean on. I can't wait to call him.
"It's been a long road but this is why you work hard. Finally got here and it feels amazing."
Snedeker had looked set to claim the opening event of the 2018-19 PGA Tour season until he made three successive bogeys from the 10th hole.
His retreat let the pack back into contention in what turned into a wild finish in conditions that started out extremely windy before abating by the final few holes.
Snedeker shot a two-over-par 74, Moore a 67 and Tway 71.
The trio finished regulation at 14-under 274, one stroke ahead of five others.
Tway birdied the final two holes of regulation, two-putting the par-five 18th to join the playoff.
After Tway and Moore twice birdied the 18th in the playoff, they continued at the par-four 10th, where Tway finally prevailed just before sunset.
Moore, who started the day seven shots behind Snedeker, was happy just to have been in contention.
"This was a bonus, just to be in this position to have a chance to win," he added.
"When I got done with my round there were plenty of guys with plenty of holes left. I was proud of how I played."
(Reporting by Andrew Both in Cary, North Carolina; Editing by Peter Rutherford)
