Stenson, trying to become only the second European to win this fickle tournament, played his best golf on the first two holes, then took advantage of some sloppy play by the defending champion on the back nine to take a lead into lunch.
The match was so wild in the early going that a hole wasn't halved until the par-3 eighth.
The desert air was frosty again, causing a 30-minute delay, and the wind came out of the opposite direction so that the first seven holes played into the chilly breeze. Stenson hit his second shot to the par-5 opening hole to 8 feet, and needing two putts for the win, made it for eagle. With a wedge in hand on No. 2, he hit his approach to 5 feet and was conceded the birdie when Ogilvy failed to get up-and-down.
But the match changed quickly, and heavily in the Australian's favor.
Stenson three-putted from 70 feet to lose the third, then Ogilvy holed a 10-foot birdie on the fourth to square the match.
Ogilvy took his first lead with as fine a 3-wood as can be struck.
Downwind on a 639-yard hole, he hit his approach from 296 yards onto the green, the ball stopping a yard beyond the hole. He never had to putt, because Stenson found the desert and had to chip out left-handed. Ogilvy then won his fourth straight hole for a 2-up lead by driving through the fairway, then hitting a wedge that one-hopped off the flag and settled 8 feet away.
Stenson won the seventh, hitting a driver through the green and down a slope, pitching delicately up the hill to 3 feet.
It set the tone for a wonderful championship match of brilliant shots and key putts, but after they finished out the front nine by halving the last two holes, it turned slightly sloppy.
Stenson won the 10th with an up-and-down from the bunker for birdie, but Ogilvy was in prime position to regain the lead at No. 11, when he found the fairway with a 3-wood, while Stenson hit into a bunker and couldn't reach the green.
Ogilvy played 30 feet left of the cup, and ran his putt down the side of a ridge some 6 feet by the hole. Stenson chipped nicely to 5 feet, and looked like he might win the hole when Ogilvy missed his par putt. But the Swede yanked his putt, too.
Stenson pulled in front with another swift change on the reachable par-4 12th. Ogilvy had the better angle into the green and pitched to 6 feet. Stenson had a far more difficult shot, and his semi-flop shot spun back against the collar, 25 feet away. Using the blade of his sand wedge, he knocked it in for birdie, and Ogilvy missed.
Ogilvy made so many putts in winning last year at La Costa and reaching the finals this year. But he began to struggle, losing another chance to square the match when he lipped out a 6-foot par putt on the 13th.
Stenson took a 2-up lead on the 15th with a 12-foot birdie putt. They halved the final three holes.