So they settled it in true gambling fashion: by going to a makeshift, 93-yard 20th hole, played from the practice putting green behind the clubhouse to a newly cut hole on the 18th green.
And they had to play that three times before Mickelson finally prevailed, holing a 4-footer to win the winner-take-all match, 1-up.
Mickelson, 48, had been talking up Friday's encounter for months and suggesting it would help him get some small measure of satisfaction if he could win after a Hall of Fame career yet still being in the shadow of Woods, who has 14 majors and 80 PGA Tour titles.
He called the victory "very special."
"I know big picture, your career is the greatest of all time," Mickelson, who has 43 career victories and five majors, said of Woods. "I've seen you do things that are just remarkable. But just know I will not ever let you live this one down. I will bring it up every time I see you."
Mickelson then referred to a champion's belt that Friday's winner received, telling Woods, "I will wear this belt buckle every time I see you."
"It's not the Masters, it's not the U.S. Open, I know, but it's something," Mickelson said. "It's nice to have a little something on you."
Woods humbly sat by, taking the good-natured trash talk he knew would be coming if he did not prevail.
"I had plenty of opportunities to make putts today, and I didn't make any putts to put a little bit of pressure on Phil. I had an opportunity on the last hole to win the match," Woods said, referring to the first playoff hole on the par-5 18th, "and I hit a bad putt. And then in extra holes -- how do you not hit the green with a lob wedge? Twice?
"So that was an opportunity that went wasted, and Phil capitalized on it."
There were indications that Mickelson was more prepared, having spent several days in recent weeks at Shadow Creek, including a good part of this week. Woods was here only for a practice round Tuesday then joined his family in Palm Springs, California, for Thanksgiving before returning Friday morning.
Both players shot 69 in the regulation holes, with Woods making six birdies and three bogeys. Mickelson made three birdies.
It appeared Mickelson would close out the match on the par-3 17th when he knocked his approach to 10 feet and Woods hit his over the green. But Woods chipped in for a birdie and Mickelson missed, squaring the match.
They both parred the 18th and then played it again, with Woods holding a clear advantage after Mickelson found a greenside bunker and had to blast to 30 feet. Facing an 8-footer to win, Woods left it low and short.
They then went to the makeshift playoff hole, where a tee was put on the putting green and measured 93 yards to a new hole cut on the 18th green. Woods twice missed the green, and after Mickelson missed a 6-footer that would have won, he told Woods to pick up his 5-footer. "I don't want to win that way,'' a gesture that Woods had invoked earlier when he had Mickelson pick up a 3½-footer on the first extra hole.
No player ever led by more than 1 hole. Woods took his only lead when he birdied the 11th to tie and then birdied the 12th to go 1-up. But Mickelson tied him with a birdie putt at the 14th then went ahead when Woods bogeyed the 15th.
Mickelson won three holes in regulation with pars.
Woods won a $200,000 side bet on the first hole when Mickelson failed to make birdie. But Mickelson won three side bets totaling $600,000 by hitting shots closer to the pin at the fifth, eighth and 13th holes.
Those side bets were to be paid to the respective players' charitable foundations with the players' own funds.
"We were there to play head-to-head and do something that's never been done before," Woods said. "Lo and behold, here we are going to extra holes and we're under the lights. You couldn't have made this event any better than it was.
"We were going back and forth. Phil had the lead, I had the lead. It flipped, and next thing you know, we go to the last hole. It was back and forth, very competitive, on a golf course that was playing a little bit on the tricky side."