The Italian was the epitome of calm on an action-packed final round, starting his day with 13 successive pars – his putter working superbly on the green.
A birdie at the 14th put him in front on his own for the first time, although Xander Schauffele quickly joined him on seven under to set up a thrilling finale.
Crucially for Molinari he produced some magic on the last, his approach setting up another birdie putt that proved decisive, signing for a final-round 69 that left him on eight under - ultimately finishing two clear of Schauffele, Justin Rose, Rory McIlroy and Kevin Kisner.
After a 30-minute wait the 35-year-old was crowned champion golfer, and that will take some getting used to.
“It is absolutely amazing,” he told Sky Sports. “It will take a long time to sink in.
“It’s been a great week. The course bit me a few times the first couple of days, but then to go bogey free over the weekend on a course like this is incredible.
“I was [as calm] as much as you can be on the final round of The Open. But for the first time I felt like I was ready for it.
“Playing with Tiger [Woods] makes it even more special. I couldn’t have written it any better. It’s been a long journey, and holding this trophy changes a few things.”
In his winners’ speech, Molinari paid tribute to his rivals and the staff at Carnoustie for creating a memorable week.
He added: “What a week! Congratulations to the runners-up. It has been a tough fight. There is only one winner unfortunately in golf, this time it is me but congratulations to them.
“The golf course was incredible, [and] this truly is the best atmosphere in golf.”
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