The 35-year-old from Turin underlined his status as the hottest player in world golf by finishing with an eight-under total of 276.
The red-hot European made par for the first 13 holes on the final day as he watched the weather play havoc with a number of tournament favourites.
Molinari saw Jordan Spieth, Tiger Woods and Xander Schauffele fall through the field as he effortlessly mastered the Carnoustie Links.
The Italian birdied his second of the back nine at the last to make it a blemish-free card of 69.
Molinari ended the week at -8, and celebrated accordingly, pumping the air with his fist and allowing a huge smile of relief and excitement to play across his face.
The eventual winner played alongside former world No 1 Woods in the final round, who appeared ready to end his 10-year victory drought in majors midway through the round.
The American topped the leaderboard until his hopes nosedived following a double-bogey at the 11th and another dropped stroke at the next hole.
The 14-times major winner had to settle for a closing 71 and a 279 aggregate.
“The way Francesco played today was beautiful,” said Woods.
Speaking after his win, Molinari thanked his team for their hard work.
“What a week!” he told the crowd.
“Well it’s incredible to stand here between all these people.
“A lot of people I should thank – my family for their support, my wife, who is a really really big part of my team.
“And of course all of my family back in Italy.
“I have such a big team behind me, each one of them has been a massive part of the journey, so thank you for all your work.
“Congratulations to the runners up, it’s been a tough fight.
“Unfortunately in golf there’s only one winner, this time it’s me! But they played very well, and congratulations to them.”
The Italian has hit a purple patch in his career so far this year, finishing in the top two of four of his last five tournaments.
Molinari, when hot favourite at the Italian Open, almost 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.
And a top 25 finish at the US Open was followed up by a win at the Quicken Loans then a runner up finish at the John Deere last week.