Francesco Molinari won the British Open, closing with a bogey-free round of 69 to finish at 8 under par.
The Italian didn't have a bogey across the entire weekend and make a crucial birdie putt at the 18th to take the outright lead.
The 14-times major champion began the day four shots adrift of fellow Americans Jordan Spieth, Xander Schauffele and Kevin Kisner but with a warm westerly breeze playing havoc, the 42-year-old mounted a thrilling early charge.
Molinari tapped in for birdie at the par-5 No. 14 to break out of a five-way tie for the lead, which included Rory McIlroy after his eagle at the same hole.
He got one back on the par five-14th but his playing partner - and eventual champion - Francesco Molinari also birdied to take him two shots ahead of the American.
However he then fell back with a double bogey at the 11th hole, leaving him one shot off the lead.
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For Woods, a return to championship form would take on special meaning, given the personal demons and injuries he has overcome. A bogey on the 16th and a par save from short of the Barry Burn on the 18th gave him a 66. "I'm right there", he said.
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He then missed a great chance to hit the front on his own at the 5th, pushing a straightforward six-foot putt to the right of pin. This was his best performance in a major since he made the same score in the second round at Augusta seven years ago.
Molinari had started the day three shots behind a trio of overnight leaders in Schauffele, Kisner and Spieth, who were all nine under par when they teed off.
But in his traditional last-day red shirt he continued where he left off the previous day to pile pressure on those above him. "I've got a chance at this, which is great".
He laid up, then speared a majestic 150-yard wedge to within three feet. Like 11, he made a decision to go aggressive, attempting a high-risk flop shot over a bunker instead of playing safely past the pin and taking double bogey out of play. At one point during the third round, Woods was tied for the lead, but he would eventually be overtaken by a few golfers who had not teed off yet.
Woods, who had six birdies in his round of 66, is another stroke back alongside Rory McIlroy (70), Webb Simpson (67), Alex Noren (67), Matt Kuchar (70), Tommy Fleetwood (71) and Zach Johnson (72).
Woods has been plagued by knee and back injuries and dealt with personal problems including a marriage breakup since winning the last of his 14 majors a decade ago.
Further trouble and another dropped shot was to follow at the next and it was with his head bowed deep in thought he strode down the par-three 13th where he managed to stem the haemorrhage with a par.
Woods' second shot on the second hole had drifted into the hay beside the fairway, and he faced the choice of either punching out to the fairway or attempting to clear the precious British heads of the gallery.