The American world number six triumphed at the first extra hole after Davis was assessed a two-stroke penalty for making contact with a loose impediment in a greenside hazard when playing his third shot.
On his backswing, Englishman Davis clipped a dry reed with his club before splashing out to 30 feet and he immediately called over a rules official to clarify the situation.
Furyk coolly sank his par putt to complete his second victory on the U.S. circuit in five weeks.
"To have the tournament come down that way is definitely not the way I want to win," the 39-year-old American told reporters. "It's obviously a tough loss for him, and I respect and admire what he did.
Davis, who had been seeking his first PGA Tour victory, had a bad feeling the minute he completed his third shot in the playoff.
"It was one of those things I thought I saw movement out of the corner of my eye," the 35-year-old Englishman said.
"I thought we'd check on TV, and indeed there was movement."
The pair finished the 72 regulation holes on 13-under-par 271, Davis rolling in a 17-foot birdie putt on the 18th green for a three-under-par 68 and Furyk closing with a 69.
A final-round shootout appeared to be on the cards with Furyk holding a slender one-stroke advantage overnight and 12 players bunched within three strokes of his lead.
Although Furyk was caught early on by playing partner Davis and American Heath Slocum, he regained a one-shot cushion after reaching the turn in three-under 33.
Davis pulled his seven-iron approach left of the green, his ball bouncing off rocks and ending up on compacted sand in a hazard.
Furyk responded by hitting his six-iron on to the fringe at the back of the green from where he putted to around six feet.
After pondering whether to take a penalty drop, Davis finally decided to play his third shot out of the hazard and television replays of the reed being struck on his backswing ultimately sealed his fate.
"I gave myself a chance the last six or seven holes, unfortunately I didn't finish it off," he said. "I'm proud of what I did on 18 the first time around, but obviously it doesn't relate to a win so it's a bit disappointing."
(Writing by Mark Lamport-Stokes in Los Angeles; Editing by Greg Stutchbury)