Molinari stormed into contention with two eagles in his last seven holes to set the clubhouse target of nine under par, which Dunne was able to match with a birdie on the last.
The players returned to the 18th for sudden death and a scrappy par was good enough to give Molinari the win after Dunne found trouble off the tee and failed to get up and down from a greenside bunker.
Dunne began the day with a two-shot lead and was three clear after recovering from a bogey on the first with birdies on the second and fifth, but bogeyed the sixth and was never able to shake off the chasing pack.
The 24-year-old, who shared the 54-hole lead in the 2015 Open at St Andrews while still an amateur, carded two more birdies on the back nine but immediately followed each with a bogey.
Another dropped shot on the 16th left him needing to birdie one of the final two holes to catch Molinari and a superb chip from the edge of the 18th green set up a tap-in for a closing 72.
After starting the day four shots off the lead, Molinari covered the front nine in level par before following a birdie on the 11th with an eagle on the par-five 12th after a brilliant approach from 250 yards.
The 36-year-old's chance looked to have gone when he bogeyed the 16th after finding two bunkers on the tough par four, but he birdied the 17th from close range and holed from 20 feet for another eagle on the last in a superb 68.