"It's been a few years since I've felt like this", an ebullient Woods said after his best score in a Major since another 66 in the second round of the 2011 Masters. "Who hasn't?" said Spieth.
Another American, Kevin Chappell, is alone in fourth place, two shots behind after a 67.
After seven smooth holes, Woods began to scramble.
The red-hot European made par for the first 13 holes on the final day as he watched the weather play havoc with a number of tournament favourites.
Spieth was at 9-under 204.
He then made a two-putt birdie at the lengthy par-five sixth hole to get two two-under. Carnoustie's first few holes are supposed to offer a warm welcome before its brutal finish, but several of them played into the wind Sunday.
Woods found the sand on the par three eighth but got it to puttable range and parred it and again on the ninth he found sand but repeated the escape act with a par and found himself alone in the lead.
In scenes reminiscent of those in the prime of his career, waves of excitement rippled through the massive gallery as 14-times Major champion Woods charged up the leaderboard.
Brooks Koepka is unlikely to be celebrating back-to-back major victories at the British Open on Sunday.
Eddie Pepperell says he was a "little hung over" during his round of 4-under 67 that has left him one shot off the lead at the British Open.
Koepka is 3 over for the tournament, 10 shots off the lead. He then got a bogey on the 12th to run out of contention. He is even par overall.
Florida sheriff won't charge man in deadly "stand your ground" shooting
Onlooker Michael Drejka got out of his parked auto and began "harassing" her about being parked in a handicap space, she said. The men were allegedly arguing about a handicap parking space when the shooting occurred, according to Newsweek .
"Playing with Tiger makes it even more special". McIlroy was within two shots of the lead until bogeys on two of his last three holes for a 70 left him four behind.
Tiger Woods shot 71 and was 5 under overall. Woods almost reached the green in two on the par-5 sixth hole that still bears Ben Hogan's name. "If I could just keep hitting the ball the way I did today and putt a little bit better, tomorrow will be a better day", he added. There already was a 64 on the board by Justin Rose and a 65 by Yusaku Miyazato.
Spieth maintained that being put on the clock at the start of the back nine really hurt his chances.
For 20 minutes on a warm Saturday afternoon at the British Open, Tiger Woods's name was back atop a leaderboard at a major championship and Carnoustie was buzzing.
Zach Johnson and Kevin Kisner are more than just housemates this week.
Francesco Molinari went into today tied for 29th - he's now tied 5th on -6 after shooting that score today.
The Adidas athlete was last year's Rookie of the Year on the tour and made a statement today, finishing just behind Molinari. He didn't make a bogey on the normally tough Carnoustie links.
His tee shot went into the crowd and, despite a fortunate bounce, his next shot over a bunker went off the back of the green.
The first to congratulate him was Woods.
Then, on the short par-four 11th, Woods struck a handsome tee shot and got up-and-down from just left of the green, using the putter twice. The clincher was a bold drive on the 18th hole that flirted with edge of a pot bunker, a wedge to 5 feet and a birdie putt that made him a major champion.
Kisner had the 54-hole lead at the PGA Championship a year ago.
The chances of the Open champion sporting a mullet after Tommy Fleetwood and Pat Perez fell from one shot off the pace to four and seven shots behind respectively.