Jordan Spieth will take a share of the lead into the final round of the 147th Open Championship as he looks to become the first back-to-back victor for a decade.
Molinari's 2-under-par 69 earned him his first major championship.
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.
Woods shot 71 on Sunday, including a back-nine 37, to finish in a three-way tie for sixth at 5-under 279 (71-71-66-71). Amidst a bogey barrage from other contenders, Molinari's round of 16 pars and two birdies helped him rise to the top.
The pair of course had some history - Molinari beat Woods with the famous concession on the final hole of the 2012 Ryder Cup at Medinah, the completion of Europe's famous comeback victory over the USA.
Molinari went one better, his second setting up an easy birdie and laying down a marker at eight under for the four players left on course.
"The only thing they've seen is my struggles and the pain I was going through".
Molinari assumed the lead at 7 under on the 14th hole with a two-putt birdie, and Schauffele tied him with a like birdie two groups hence.
Naturally all the attention was focused on Tiger, and then when he made two bad mistakes on the 11th it fell on Jordan Spieth and Schauffele as they at first looked solid and then unravelled, and then on Kevin Kisner and Kevin Chappell before they faltered. "I put good swings on it".
Tiger Woods of the U.S. walks on to the 18th green during the final round
"Momentum-wise, birdieing the very first hole, picking up where I left off, got me well into the round".
One of those runners-up finishes came after blowing a five-shot lead with nine holes to play in defence of his Masters title, while the fourth place came when he finished a shot outside a three-man play-off for the 2015 Open at St Andrews in pursuit of the third leg of an unprecedented calendar grand slam.
"I've played it out at Augusta in my head".
Defending champion Spieth failed to make a birdie, with a penalty drop from a gorse bush on the sixth leading to a double bogey in a final round of 76. "I need to go get it", Woods said.
Francesco Molinari's win at the British Open has broken the American stranglehold on golf's major championships, with a USA golfer having won the previous five.
"I think it will take a long time for this to sink in".
It was just as tough for Thomas, who bogeyed two of the last three holes to miss the cut by a shot.
"The way that Francesco played today was lovely", said Woods.
"I was in the final group, I had to face a little bit of adversity early in the round, and I still gave myself a chance". However, it proved to be a false dawn as three dropped shots in two holes saw him fall behind and in the end the 14-time major victor was being consoled by his two children.
The one shot Woods would like to have back was the 13th, where he made a 15-foot putt for par.
Soon after Rose had finished, McIlroy made his way down the final hole, but his second shot was 20 feet from the pin and he two-putted to remain six under. I missed a few shots, but I was up-and-down from pretty well from everywhere.