Molinari, who had two wins and two second places in his previous five starts this season, carded a nerveless closing 69 at a windswept Carnoustie to finish eight under par, two shots clear of Justin Rose, Rory McIlroy, Kevin Kisner and Xander Schauffele.
Playing alongside a rejuvenated Tiger Woods, Molinari followed 13 straight pars with a birdie on the 14th and finished a brilliant round in fitting style with another from just three feet on the 18th.
Rose, who had birdied the 18th in the second round just to make the cut, followed his record-equalling 64 on Saturday with a 69, while McIlroy recovered from a slow start with the aid of an eagle on the 14th to record his third straight top five in the Open.
Defending champion Jordan Spieth went into the final round tied for the lead with Schauffele and Kevin Kisner, but stumbled to the turn in 39 and Kisner managed one shot worse to throw the tournament wide open.
For much of the afternoon it looked as though everything was falling into place for Woods to pull off a barely believable victory, his first of any description since 2013, the 15th major of his amazing career and a first since the 2008 US Open.
Birdies at the fourth and sixth took Woods to the turn in 34 and, with the leaders crumbling, gave the 42-year-old a one-shot lead over Molinari and Spieth, who had double-bogeyed the sixth after hitting his second shot into a gorse bush.
But just when one of the greatest sporting comebacks of all time seemed on the cards, Woods failed to pull off a risky flop shot to get out of trouble on the 11th and ran up a deflating double bogey.
Another dropped shot on the next effectively ended his chances but the drama was just beginning elsewhere, McIlroy following birdies on the ninth and 11th with an eagle from 35 feet on the 14th.
That vaulted McIlroy into a six-way tie for the lead and Rose joined the expanding group on six under with a tap-in birdie on the 18th, his fourth in succession on the closing hole. His second on Friday had been required just to make the cut on the mark of three over par.
Molinari broke out of the pack with his first birdie of the day on the par-five 14th and up ahead McIlroy had to settle for joining Rose in the clubhouse lead after failing to birdie the last.
Schauffele, who had struggled to the turn in 40, birdied the 10th and 14th to join Molinari in the lead, but bogeyed the 17th and was unable to find the magical finish required on the last to deny Molinari a deserved victory.
Spieth’s bid to become the first back-to-back Open champion since Padraig Harrington in 2008 came to a miserable end with a closing 76 to finish four shots off the pace.