Molinari, 35, produced a faultless final round to shoot a two-under 69 in windy conditions on the Scottish links.
He finished two ahead of four players, including home favourites Justin Rose and Rory McIlroy, tied on six under.
Tiger Woods finished five under after leading midway through the round, with Jordan Spieth on four under.
Molinari emerged from a six-way tie for the lead, when all the players were either on the back nine or in the clubhouse, to lift the Claret Jug.
The Europe Ryder Cup player started the final day three shots adrift of overnight leaders Spieth, Kevin Kisner and Xander Schauffele, but maintained composure while his rivals all dropped shots in a thrilling battle for the lead.
Molinari, ranked 15th in the world, had been tipped by a number of experts before the tournament as a player to watch at Carnoustie – and he proved them right with an assured display over the four days.
Despite a modest Open record, he had been backed on the strength of a fine few months which saw him claim the PGA Championship at Wentworth – his biggest win on the European Tour – and his maiden PGA Tour victory at the Quicken Loans Classic.
Steady opening rounds of 70 and 72 took him into the weekend, before a stunning 65 on Saturday propelled him into contention.
Molinari stressed the importance of making pars on Sunday in an interview after his third round and fulfilled his target by beginning with 13 in succession in testing conditions.
That steady sequence was ended with a birdie on the par-five 14th, before a glorious approach on the 18th allowed him to convert another.
Molinari looked anxious as he waited in the scorer’s cabin, watching nearest challenger Schauffele close his round before moving outside to the putting green when the American bogeyed the 17th.
Schauffele, 24, needed to eagle the last hole to force a play-off, his approach landing short and allowing Molinari to celebrate with his caddie and wife.