Englishman Rose watched in amazement as his ball dropped into the cup after breaking sharply from left to right downhill to cap a four-under-par 66 at East Lake Golf Club.
He led the 30-man field for much of the day before Woods, playing in the final pairing with FedExCup points leader Rory McIlroy, caught him with two birdies in the last four holes.
Woods narrowly missed a birdie attempt from 11 feet on the 18th green, his ball sliding past the left edge, and had to settle for an opening 66 in the US PGA Tour's fourth and final play-off event.
"I played well today," Woods said after mixing six birdies with two bogeys on a long and tight East Lake layout where the greens ran lightning fast.
"I hit a lot of good shots out there. I made a couple of putts out there, but more importantly, I got up and down. I was almost right on my number what I could have gotten out of it today."
Americans Steve Stricker, Matt Kuchar, Bo Van Pelt and Scott Piercy carded 67s while world number one McIlroy, bidding for a fifth US PGA Tour victory this year, and his fourth in just five starts, returned a 69.
"If you don't hit fairways, it's hard," Northern Irishman McIlroy said after hitting only seven of 14 fairways. "If you hit the ball in the rough here, it's very, very difficult to get any control on your ball.
"But I'm in a good position going into tomorrow. I'll just go out there tomorrow and give it my best again like I did today."
Rose, who clinched his fourth US PGA Tour victory at the WGC-Cadillac Championship in March, was delighted to limit potential damage arising from erratic form off the tee with a sizzling putting display.
"You've got to keep a much bigger premium of hitting it on the fairway here, and I didn't do a great job of that today," Rose said after missing seven of 14 fairways while totalling only 26 putts.
"That's something I can improve on in the next few rounds. If I keep it on the fairway, I feel my iron game and short game is coming around.
"I'm beginning to get some confidence with the putter, so it was nice to get out of today with a hot start, considering that."
For Rose to land the season-long FedExCup and its $10 million bonus, he knows he is a long shot who has to win the Tour Championship and hope that players ranked higher in the points standings finish poorly.
Rose was 24th at the start of the week while any of the top five - McIlroy, Woods, Nick Watney, Phil Mickelson and Brandt Snedeker - would automatically secure the trophy with victory on Sunday.
"Just being here at this tournament is a good feeling," said Rose. "Obviously, I'd like to be much higher up the rankings.
"Last year I didn't factor in the tournament and this year maybe being a little bit looser out there is going to help me.
"I've only got this tournament on my mind. A lot of the other guys have two trophies on their minds. I think I'll steal it if I end up winning both of them."