Woods, now 42 and without a major win since 2008 as personal problems and health issues have stalled his career, is playing with Italy's Francesco Molinari, who was the best-placed European at six under par.
Woods held the outright lead after 10 holes but shot a double-bogey on No. 11, followed by a bogey on No. 12.
But that was a bridge too far for the man who was the co-leader overnight, as the Claret Jug was secured for the in-form Molinari.
Jordan Spieth, the defending champion and three-time major victor by age 24, began Sunday with a share of the British Open lead, pinned together with Kevin Kisner and Xander Schauffele, two ahead of the next-closest challengers.
Meanwhile, Fleetwood is the most likely to become the first English victor of the Claret Jug since Nick Faldo in 1992. Schauffele's hopes vanished with a bogey on the penultimate hole, meaning an eagle was required on 18 to force a play-off.
However, Tommy Fleetwood's chances of becoming the first English victor since Nick Faldo in 1992 looked to have gone after he pulled his third shot on the sixth out of bounds, the resulting double bogey leaving him six off the pace. "I got off to not exactly the best start, being 2 over through three, but got it back", he said. All day long they had heard the cheers rolling across the links of Carnoustie and, finally, he was here.
He fell to 5 under, a stroke behind four players heading into the tough back nine.
Tiger Woods sent a buzz throughout the sports world and turned the clock back when he briefly led the British Open on Sunday, but a double bogey-bogey stretch on the back nine ultimately doomed his quest for major number 15.
Spieth's victories would be the other way around and make it four wins in his past 15 majors, during which time the 24-year-old has also finished second twice, third and fourth.
Even defending champion Jordan Spieth seemed ordinary, in a round where he was doing some extraordinary things.
"Just like the ones I've practiced in my backyard", he said, adding, "That was big for me just to not finish with two bogeys on the last three holes, playing as well as I did". He missed a short birdie putt and tapped in for par to finish off his final-round 71.
Only four of the first 60 players to complete their rounds on Sunday carded a sub-70 round.
The yellow grass of a dry summer in Scotland comes to life Sunday because of a red shirt at the British Open. It's the closest he's been to the lead in a major since five years ago at Muirfield. The American began four shots off the lead, but went out in 34 to claw his way to the top.
For much of the afternoon it looked as though everything was falling into place for Woods to pull off a barely believable victory, his first of any description since 2013, the 15th major of his awesome career and a first since the 2008 US Open.