The 35-year-old Italian went one better than compatriot Costantino Rocca, who lost to John Daly in a play-off at St Andrews in 1995, after clinching a two-shot success on the Angus coast following a closing 69 for an eight-under-par 276 total.
McIlroy, the 2014 winner, finished joint-second alongside Justin Rose and American duo Kevin Kisner and Xander Schauffele, with 14-time major champion Woods a further shot back along with compatriot Kevin Chappell and Englishman Eddie Pepperell.
Molinari had gone into the season’s third major on the crest of a wave, having recorded wins on both the European Tour and PGA Tour, as well as two second-place finishes, over the past couple of months.
But, despite that hot run of form, the London-based player didn’t fancy his chances here due to the fact he had never played well at Carnoustie in the Dunhill Links Championship. In seven appearances in the pro-am event, he had not managed a top-30 finish and, partly because of that, hasn’t played in it since 2014.
“I’m lost for words, really,” admitted Molinari. “It is incredible to do something like this and I am very proud of what I have done. It’s amazing to sit with the Claret Jug.
“I knew I was coming in with some good golf, but my record around here was terrible. So that didn’t make me too optimistic about the week, but I just tried to not think about it and focus on hitting good shots day by day.
“To be completely honest, it’s one of the reasons why I didn’t play the Dunhill Links in the last few years because I got beaten up around here a few times already in the past. I didn’t particularly enjoy that feeling. It’s a really tough course. You can’t really hide.”
Playing with Molinari, Woods held the outright lead on seven-under as he played the tenth hole only to see his hopes of ending a major drought stretching back to 2008 disappear as he ran up a double-bogey 6 at the 11th. The former world No.1, who has had chronic back trouble in recent years and feared his career may have been over at one point last year, finished joint-sixth, three behind the winner.
Asked how ruthless he’d had to be to kill off what would have been a fairytale win for Woods, Molinari said: “I don’t think it’s about being ruthless. Obviously, it’s golf. It’s sports. We’re competing against each other.
“But I was competing against all the other guys as well, not only against him. There was very, very good sportsmanship today during the whole round. Obviously, he had it going on the front nine, and then, unfortunately, just hit a couple of loose shots on the back nine. It’s hard not to pay for missed shots around Carnoustie.”
This triumph came in Molinari’s tenth appearance in the event, with his best previous performance – joint-ninth – coming on a similar fast-running course at Muirfield in 2013. He played with Phil Mickelson on the final day then and used that experience to good effect when his chance to become a Claret Jug winner arrived.
“I got here only Monday lunchtime from the States and walked a few holes, and I saw it was firm and fast and the rough wasn’t too bad. And it reminded me of Muirfield,” he admitted. “Obviously, that was a great experience to play with Phil in the last round and to see someone doing the job, getting the job done on Sunday.
“So I liked the way the course was playing, but, again, it’s a beast of a course. So I don’t think anyone feels too confident when they stand on that first tee at Carnoustie. I’m lost for words really. Incredible to do something like this, and very proud of what I’ve done.”
The win left him having to change his travel plans, revealing of a scheduled journey home last night from Edinburgh to Gatwick: “I had an easyJet flight at 9pm, so I think that’s gone!”
After a disappointing 76, defending champion Jordan Spieth slipped from joint-leader at the start of the day to a share of ninth along with Matt Kuchar and Tony Finau.