The usually low-key Italian raised his arm in the air to hail his final putt, the longest he has holed on the PGA Tour this year.
Molinari posted a 12-under 276 total and, after a two-hour wait in the clubhouse, celebrated his third PGA Tour victory when none of the overnight frontrunners could match his total.
Overnight leader Matthew Fitzpatrick shot 71 to claim second place on 10-under.
Defending champion Rory McIlroy carded 72 for equal sixth, four strokes behind Molinari.
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"I thought there was an outside chance and just started making putts," 36-year-old Molinari told NBC television about his mindset after he had teed off five strokes from the lead.
"Probably have to say my best putting round ever.
"I don't know if I could have done a lot more than that to be honest."
Asked what had elevated his game over the past year, he said confidence had a lot to do with it.
"In the summer I went through a run of golf that was hard to believe, even for me.
"The winter came and it was time to find some motivation, which wasn't extremely easy after last year. The smart thing to do was to take a long break."
Molinari's round tallied eight birdies, including a 15-yard pitch-in at the eighth and his 64 was the lowest round all week at the tournament.
England's Fitzpatrick came closest to catching Molinari, but rued a missed 12-foot eagle putt at the 16th that would have cut the deficit to one stroke.
He held on to finish a stroke ahead of compatriot Tommy Fleetwood (68), Spaniard Rafa Cabrera Bello (69) and South Korean Im Sung-jae (69).
While Im, compatriot Kang Sung-hoon and American Keith Mitchell earned invitations to the British Open, as the top three finishers not already exempt the day belonged to Molinari.
"It's high up there with the best wins I've had," Molinari said, describing Palmer, who died in 2016, as a "global icon" of the sport.
"Coming from Italy, he and Jack (Nicklaus) were up there as gods."
(Reporting by Andrew Both in Cary, North Carolina; Editing by Ken Ferris)
