In glorious sunshine with barely a breath of wind at East Lake Golf Club in the US PGA Tour's season finale, Stenson piled up five birdies in six holes with a superb exhibition of pinpoint approach play on the way to a six-under-par 64.
Though he dropped his only shot of the day at the par-four 16th, where he found a poor lie in the right rough off the tee and missed the green with his approach, he signed off in style with a four-foot birdie putt at the last.
"When it's your day, you're hitting good shots and getting them as close as I got them," Stenson, ranked second in the FedExCup standings, said after mixing seven birdies with a lone bogey on the long, challenging East Lake layout.
"It's a golf course that it's hard to play well out of the rough, that's for sure, and I definitely hit a lot of fairways and greens.
"Just a very good round of golf but it's still very early in the tournament. One day down, got another three to go."
It was not a day to remember, though, for FedExCup points leader Woods who failed to register a single birdie while making three bogeys on the way to a 73, ending the round in 29th place with only USPGA Championship winner Jason Dufner (74) below him.
It was the first time Woods had completed a round on the PGA Tour without a birdie since his opening round at the 2010 US Open, and only the eighth time during his career on the US circuit.
Masters champion Adam Scott of Australia, who reeled off six birdies in seven holes from the 10th, was alone in second after firing a 65, finishing a stroke in front of Americans Billy Horschel and Steve Stricker.
Another American, Roberto Castro, opened with a 67 at East Lake where FedExCup play-off honours and the eye-popping $10 million bonus are also on the line, offering huge added incentive to the 30-man field.
US Open champion Justin Rose of England, South African Charl Schwartzel, Spaniard Sergio Garcia and Australian Jason Day were among a group of eight players knotted on 68.
Woods, seeking a sixth USTour victory this year, missed a six-foot birdie putt at the par-four first, then bogeyed the tricky par-four fifth after hitting his approach into a greenside bunker to reach the turn in one over.
He also dropped shots at the 10th, where his tee shot sailed way right, and at the 14th, where he missed the fairway to the left and overshot the green with his second, to slide further down the leaderboard.
Woods only hit eight of 14 fairways and 12 greens in regulation, while totalling 34 putts after missing four birdie attempts from inside 10 feet.
"We know he didn't have the best of days, and he's going to fight hard to try to come back into the tournament," Stenson said of the world number one.
"It's still a long way to go, but it's always nice to perform the way I did when you're playing with the world's best player."
While Woods struggled, the Swede commanded the spotlight after his electrifying start, despite having come into this event with concern over tendinitis in his left wrist which developed during last week's BMW Championship in Chicago.
He launched his birdie blitz by hitting his tee shot to six feet at the par-three second and knocking in the putt, then picked up another stroke at the fourth where his approach finished just eight inches from the cup.
Stenson, who won the Deutsche Bank Championship earlier this month, struck his second shot to four feet at the fifth, his tee shot to within two feet at the sixth and sank his longest birdie putt of the round at the seventh, a nine-footer.
"First seven holes is probably as good a proximity to the hole as I've ever had on seven iron shots and I made five birdies out of that," Stenson said.
Out in a sizzling five-under 30, Stenson also birdied the 15th before offsetting his bogey on 16 with a birdie at the last.
The Swede was also delighted his tendinitis appeared to be on the mend.
"My wrist didn't feel bad this morning when I woke up," he said. "It's been a pretty heavy regime there, icing it, a couple of anti-inflammatories and treatment on it.
"That seemed to have done the trick. We're happy with that, and I hope I continue the recovery."
Stenson put himself in prime position to land FedExCup honours as only he and the other members of the top five in the standings - Woods, Scott, Zach Johnson and Matt Kuchar - would automatically secure the play-off trophy with victory on Sunday.
- Sports & Recreation