Molinari, in Justin Thomas' words, is arguably "the hottest player on the planet", having recorded three wins in his past six worldwide starts.
"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.
"This was a special moment for Francesco but it was also a special moment for European golf", he told Sky Sports.
And Molinari - who is the first Italian to win a major was delighted to have won at Carnoustie.
"But 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 in 2013, when I played the last round with Phil (Mickelson)".
"I just hope Italy, which is not a large golfing country, will realise what he did, 1.4 billion people would eventually see The Open and see this man from Italy win. So there's going to be a lot of European guys vying for his partnership in foursomes at the Ryder Cup". "He's a fantastic golfer and he's a great guy".
Woods, Rory McIlory and Jordan Spieth all held at least a share of the lead on Sunday, before Molinari won it by two shots. "You know, everything", McIlroy said.
"It's a different version".
No regrets for McIlroy after Open second
"But he [was] right there".
"He's getting himself in the mix". He looked good in D.C. a couple weeks back (at the Quicken Loans).
McIlroy slipped back to three under early in his round but powered back into contention on a congested leaderboard, which included a rejuvenated Tiger Woods, with an eagle on the 14th. He played so good all week.
The prize for second place was set at £824,000 pre-tournament but as there were four players tied for second, this is reduced.
The 29-year-old Northern Irishman looked down and out in the closing round when bogeys at the second and fifth pushed him down the field in stiff breezes at Carnoustie that gusted up to 28mph.
He had briefly tied the lead when he got to eight under after holing a 20-foot putt for his second birdie of the day at the second hole, but a late bogey left the four-time major victor three adrift of halfway leader Brian Harman.
Molinari's win broke a run of four Americans winning the major championships, Brooks Koepka twice (2017 & 2018 US Open), Justin Thomas (2017 PGA Championship) and Patrick Reed winning the Masters.
"It was great to get the crowd behind me".
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