Molinari, who had two wins and two second places in his previous five starts this season, carded a nerveless closing 69 at windswept Carnoustie to finish eight under par, two shots clear of Justin Rose, Rory McIlroy, Kevin Kisner and Xander Schauffele.
Asked how big his win would be back in Italy, Molinari joked: “It depends. If Ferrari won [in F1] they will probably get the headlines.
“It was large news this last run of form. To achieve something like this is on another level. Hopefully there were a lot of young kids watching on TV, like I was watching Costantino (Rocca) in 1995 coming so close to winning at St Andrews.”
Molinari won the BMW PGA Championship at Wentworth in May and was second in the Italian Open the following week before following a tie for 25th in the US Open with an eight-shot victory in the Quicken Loans National and second place in the Travelers Championship.
The 35-year-old only arrived in Carnoustie on Monday lunchtime and walked a few holes of a course where he missed the cut on his Open debut in 2007.
“It’s one of the reasons why I didn’t play the Dunhill Links (which uses Carnoustie as one of three courses) in the last few years because I got beaten up around here a few times already in the past,” Molinari added.
“I didn’t particularly enjoy that feeling. It’s a really tough course. You can try and play smart golf, but some shots you just have to hit it straight. There’s no way around it. You can’t really hide.
“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 go the weekend bogey free, it’s unthinkable, to be honest. Playing with Tiger [Woods] was another challenge because of the crowds and everything.
“But I felt really good [yesterday morning]. When I came here, I felt I was ready for the challenge. Obviously conscious that it could have gone either way, but I knew I was going to do my best.
“I’m lost for words really. Incredible to do something like this, and very proud of what I’ve done.”
Woods held the outright lead at Carnoustie heading into his back nine as he looked to end his decade-long major drought, but dropped three shots in two holes and never regained momentum.
The 14-time major winner eventually finished three shots behind Molinari.
However, swept along by a massive gallery cheering his every move, this was the biggest indication yet since returning from a fourth back operation that Woods can compete at the highest level again.
He admits contending at the Open felt just like the old days and was delighted his two children finally got to see what all the fuss was about.
Daughter Sam, aged 11, and son Charlie, nine, were too young to see their father in his pomp, but they greeted him with a hug behind the 18th green.
“I told them I tried, and I said, ‘hopefully you’re proud of your pops for trying as hard as I did’,” he said. “It’s pretty emotional because they gave me some pretty significant hugs and squeezes there.
“I know that they know how much this championship means to me and how much it feels good to be back playing again.
“To me it’s just so special to have them aware because I’ve won a lot of golf tournaments in my career, but they don’t remember any of them.
“The only thing they’ve seen is my struggles and the pain I was going through. Now they just want to go play soccer with me so that’s just such a great feeling.”
Joint overnight leader Jordan Spieth focused on the positives after seeing his title defence end in disappointment with a 76.
The 24-year-old said: “I’m fine. When you put yourself in position it goes your way sometimes, it doesn’t go your way sometimes.”
Schauffele insists he will look back on a chaotic final day at Carnoustie as a positive experience despite missing the chance to win his first major.
The 24-year-old American said: “It’s just going to go in the memory bank as a positive. I had a chance to win a major championship.”
Schauffele was still in contention with two holes to play, needing to pick up a couple of shots to tie the lead, but a bogey at 17 left him requiring an unlikely eagle at the 508-yard last and that was a challenge too far.