The 29-year-old carded a final-round two-under 69 to post a clubhouse score of seven-under par and he then had to sit and wait to see if any of the players following him in could match his total.
Mickelson and Bradley needed to birdie the difficult par-four 18th to force extra holes and they both holed superb putts from distance to sneak into a play-off.
But, after the trio shared pars at the first extra hole, it was Haas who prevailed, sinking a monster putt at the second play-off hole - the 10th - to land his fourth PGA Tour title.
The South Carolina resident was delighted to win the Californian event but admitted that lady luck was smiling on him during the play-off.
"I never expected to make a 40-footer, and especially in that situation," he told pgatour.com. "That was a little bit of luck involved, but felt like I put a good roll on it, and it was meant to be, I guess."
Sergio Garcia finished in a tie for fourth place on five-under par after posting a remarkable round of 64 on Sunday.
The Spaniard carded five birdies and two eagles to race up the leaderboard and put pressure on the leaders.
He could not quite finish the job off but will be pleased that his resurgence in form continues with the Accenture Match Play set to take place later this week.
Justin Rose finished in a tie for 13th on -2, while Padraig Harrington ended on four-over par after a one-under par 70 on Sunday.