Behind Woods, the leaders entering Sunday's final round were busy imploding, as reigning champion Jordan Spieth lost his magic to the tune of a 5-over 76.
Woods embraced his two children, telling them he hoped they were proud of him for trying hard.
Kisner, who had begun the day on nine-under with Spieth and Schauffele, suffered a awful start with a double bogey at the second and a bogey at the third to slip back although he responded with a chip-in birdie at the fifth.
It capped off the best stretch of golf in his career.
Woods dropped his putter in disbelief when a birdie attempt refused to drop on 13, but he made birdie at the par-five 14th to put his name at the top of the leaderboard - a development that had seasoned golf hacks snapping photos of the huge media scoreboard.
Then something unbelievable by today's standards happened: Woods teed off and started putting together a great final round, and others started falling off. Improbably, Woods found himself someplace he hadn't for quite some time.
The moment Schauffele's approach stopped short on the 18th green the celebrations could start. Schauffele made double bogey on the next hole, leaving one in the wispy grass, another over the green. You can try and play smart golf, but some shots you just have to hit it straight.
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Feeding off the electrifying atmosphere, Woods sent a laser-guided second shot to the 10th that left him a tap-in birdie, then at the 11th he rolled a 95-foot eagle putt to five feet and tapped home to gain another shot.
Molinari, when hot favourite at the Italian Open, nearly swept everyone away in the romance of it all by finishing runner up to Thorbjorn Olesen who stole a march on the final few holes.
"He hit a lot of little attractive pitches too to basically kick-ins", said Woods. Six were still tied on the back nine.
Molinari is the first Italian to win a major. You're going to get some quirky bounces, and hopefully they go your way for the week.
Two hours earlier, Woods had one hand on the Claret Jug for what would have been the fourth time after surging to the lead with a flawless front nine on a day when the previously missing wind finally bared some teeth.
The former Masters champion rued a killer double-bogey on the par-four third after his lob wedge ballooned in the breeze back into the burn and he was unable to get it up and down for bogey. Serena and I are good friends.
It was a surprise to see any one player finish clear of the pack after an afternoon of twists and turns that had left a play-off - as there had been in each of the last three Opens here - looking inevitable. 5 and 6 let a handful of contenders squeeze past Spieth, who owns three major championships, and bogeys at two of the last four holes killed his chance of catching up.
Australian Open champion Cameron Davis closed with an impressive 1-under 70 to finish in the top 40 on his major championship debut.
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