The 35-year-old became the first Italian to win a major when his two-under-par round of 69 helped him eclipse the likes of Tiger Woods, Rory McIlroy and Jordan Spieth on a nerve-jangling final day around Carnoustie.
Only 14 of the 79 players on breezy Sunday shot a sub-par round but he churned out par after par before birdies at the 14th and 18th earned him the Claret Jug.
Asked whether his triumph would knock more traditional Italian passions off the sports pages, Molinari said: "It depends. Ferrari won today? If they won, they'll probably get the headlines."
As it was Ferrari did not win and with the Italian soccer team missing the World Cup, golf might just have a chance to thrust itself out of the shadows.
"Obviously, to achieve something like this is on another level," he said.
Hopefully, there were a lot of young kids watching on TV today, like I was watching Costantino (Rocca) in '95 coming so close. Hopefully, they will get as inspired as I was at the time, watching him vie for the Claret Jug.
Rocca famously lost a playoff to American John Daly having holed an incredible putt from the so-called "Valley of Sin" in front of the 18th green at St Andrews in 1995.
It took an Italian 23 years to challenge again at the Open and this time Molinari prevailed, having managed only three top-10 finishes in his 35 previous majors.
Despite having won two of his last five tournaments in the build-up to Carnoustie, Molinari was considered an outsider.
Tiger Woods et Francesco Molinari lors du 4e tour du British Open
Image credit: Getty Images
He said the lack of spotlight on him helped - especially as he was playing alongside a charging Woods on Sunday in front of galleries 10 deep in places. While others were losing their cool, Molinari simply buckled down and cranked out a winning score.
"Clearly, in my group, the attention wasn't really on me, let's put it that way," Molinari, who rocketed from 15th to sixth in the world rankings and clinched his place in Europe's team for the Ryder Cup in France in September, said.
If someone was expecting a charge, probably they weren't expecting it from me, but it's been the same the whole of my career. I don't really care too much about it.
"I care about the people around me, the work that we put in. They know how much we've all worked to get here."
In 2010 Molinari and his elder brother Edoardo became the first brothers to compete together in the Ryder Cup since 1963.
Edoardo has since suffered with wrist and hand injuries and slumped down the rankings before a return to form last year, when he won the Trophee Hassan II in Morocco.
The new Open champion said he hoped his victory would also be a boost to his sibling with whom he won golf's World Cup for Italy in 2009.
"I would love for him to get back to where he was a few years ago," he said. "Golf is a tough beast. He's experienced some bad injuries and two hand surgeries, but he's come back.
"I'm sure this will motivate him even more to achieve some great things in the game of golf in the future."