Stenson mixed birdies and bogeys on a warm afternoon and goes into the final round with a one-shot lead for the second time in four years at Bay Hill, where he has never won.
At the par-3 second, Woods found a bunker but blasted out inches from the cup to save par but stumbled on his next swing, sending his tee shot way right and out of bounds by inches on the third hole on his way to a double-bogey six.
Tiger Woods also goes into Sunday with a shot at winning. "But everyone says that, I'm not different", he said.
More troublesome was the number of players that are still very much in range, and one of them could be Tiger Woods.
Patrick Reed, who bogeyed the 72nd hole in the Valspar Championship to miss out on a play-off with Casey, was alongside Woods on four under, with England's Justin Rose a shot behind after a remarkable fightback.
Rose's playing partner Woods also clawed his way back into the tournament with a 69 and at seven under lurks ominously. On the par-5 16th, with his ball just below the fairway bunker lip, he decided against laying up, then flew trees and water to find the green, later two-putting for another birdie. Rory McIlroy has said in recent weeks that with so much commotion, it costs Woods a half-shot per round.
This time, he has nine players in front of him, and they have some pedigree.
A local favorite, major champions, rising stars and the biggest name in golf make for an intriguing leaderboard following the first round of the Arnold Palmer Invitational.
Staying below the hole on most greens has been pivotal, Stenson said.
The 2016 Open champion is going in search of a seventh PGA Tour title, while DeChambeau is looking to land his second. Woods made the turn in 1-under 35 after a bogey on No. 8.
Stenson and DeChambeau were at 11-under 133.
He, too, missed an opportunity to pull away with Stenson. Rickie Fowler (71), Billy Horschel (70) and Luke List (67) were five shots behind.
"Today was a hard fight, it was a grind".
"I kept patient and then I got on a strong run and made three (birdies) in a row coming home". "Maybe a low one tomorrow will give me a chance to kind of steal one from behind". He has a chance to end his drought, which is all he wanted.
This Monday morning their marketing manager would have been jumping for joy such was the coverage of their sponsorship of a golf tournament at the Innisbrook Copperhead course in Tampa.
Fowler was tied for the lead briefly until a long three-putt on the par-3 14th and a messy finish.
"You look at the history of how Tiger has done well at this event and he's killed the par-fives and played really conservatively and within his comfort zone the rest of the way, and that obviously has done him pretty well".
At least he knows the course, even if he hasn't been here in five years.
Day and Hideki Matsuyama, out since Phoenix with a wrist injury, will play with Woods over the next few days. "Can't complain about anything I did today".
"That was as clean a ball-striking round as I have had all year, so I am excited".