The Italian completed a perfect haul of five points with a 4 and 2 victory over Phil Mickelson as Europe reached the 14.5 point mark.
The United States won three of the top five singles matches on Sunday to close the gap on Europe to just one point.
Europe’s 10-6 lead meant they came into day three needing just four-and-a-half points from the 12 singles clashes for a ninth Ryder Cup win in the last 12 encounters.
But it was Jim Furyk’s men who struck first on a thrilling final day, putting the first point on the board when rookie Justin Thomas beat Rory McIlroy by one hole in the top match in three-and-a-half hours.
McIlroy led for large spells of the match but things fell apart for him at the last as he sent his third shot into the water after getting into bunker trouble.
After Englishman Paul Casey took half a point from his match against three-time Major Champion Brooks Koepka to put Europe 10.5-7.5 up, Webb Simpson completed a 3 and 2 win against Justin Rose to reduce America’s deficit to two points.
Tony Finau then ended his fellow rookie Tommy Fleetwood’s 100 per cent record with a thumping 6 and 4 win in match five to make it 10.5-9.5.
Ian Poulter birdied the last to secure a 2 up win over World Number One Dustin Johnson, and with Henrik Stenson, Molinari and Sergio Garcia all dormie in their matches, victory was assured.
Mickelson then found water off the 16th tee, and with Open Champion Molinari in close, the pair shook hands.
Stenson duly completed a 5 and 4 victory over Bubba Watson, Garcia became Europe’s all-time leading point scorer with a 2 and 1 triumph over Rickie Fowler and, although Patrick Reed secured a 3 and 2 win over Tyrrell Hatton, Alex Noren won his match with Bryson DeChambeau to ensure the final scoreline finished 17.5-11.5.