By Mark Lamport-Stokes
The 29-year-old Haas sank a curling 43-foot birdie putt at the second extra hole, the driveable par-four 10th, to seal victory in bright afternoon sunshine at Riviera Country Club.
Mickelson, seeking a second consecutive PGA Tour win, missed his birdie chance from the back right bunker while Bradley's attempt, a 15-foot putt, slipped past the right edge of the cup.
Bradley, winner of last year's PGA Championship, had set up his birdie opportunity by splashing out of the front right greenside bunker, a shot he described as his best all week.
"I couldn't have hit a better shot," Bradley told reporters. "I had about a 15 percent chance of keeping that on grass (the green) where I could putt it. It was one of the best shots I've ever hit.
"But Bill Haas, you've got to give him a lot of credit. It's good to see a fellow belly putter make a putt like that. I'm happy for him."
The trio had finished the 72 regulation holes on seven-under-par 277, Haas closing with a two-under 69 while Mickelson and Bradley each birdied the par-four last for matching 71s, Mickelson holing a 26-footer and Bradley following suit with a 13-footer.
Spaniard Sergio Garcia fired a best-of-the-week 64 that included two eagles to share fourth place at five-under with Americans Dustin Johnson (71) and Jimmy Walker (69), and Australian Jarrod Lyle (70).
Joint pacesetter overnight with Bradley, Mickelson began the final round by matching his playing partner with a one-putt birdie at the par-five first to remain in a two-way tie at the top.
However, Bradley bogeyed the second and fourth to slide backwards before Mickelson sank a four-footer to birdie the fifth and move two strokes clear of the chasing pack at nine-under.
Mickelson then stumbled with bogeys at the eighth, where he missed the green to the left with his approach, and the ninth, where he found a bunker off the tee.
That put him one shot behind Haas, who birdied the driveable par-four 10th to lead by one.
Haas, last year's FedExCup champion, dropped back with consecutive bogeys from the 12th before Mickelson and Bradley regained a two-way share of the lead at eight-under when they each birdied the par-five 11th.
Though Bradley faltered with a bogey at the 13th where he missed the green to the left with his approach, Mickelson joined him at seven-under after a three-putt bogey at the 14th.
Haas responded with a two-putt birdie at the par-five 17th and moments later found himself with the outright lead after Bradley and Mickelson each bogeyed the difficult par-four 15th.
However, Bradley and Mickelson, playing together in the final grouping, then sank their birdie putts on the 18th green amid deafening roars from the crowd to take the tournament into a playoff.
(Reporting by Mark Lamport-Stokes; Editing by Ian Ransom)