Ian Poulter's dramatic 2-up win against Dustin Johnson had sent Thomas Bjørn's hosts to a guaranteed 14 points with Henrik Stenson 5-up on Bubba Watson with five to play, before Molinari finished the job.
“It's been an incredible week,” said Molinari. “It's been a long week, playing five matches, but you know, again, I
stepped it up when I had to and you know, that putt on 14 was great. Just amazing.”
Jon Rahm had earlier dealt Tiger Woods a fourth loss of the Ryder Cup as Europe weathered an early fightback from the US at Le Golf National.
Spanish rookie Rahm won his first Ryder Cup point with a 2 and 1 victory over Woods, the American suffering just his second singles defeat in eight matches in the competition.
“To beat Tiger, one of the greatest, if not the greatest, on Sunday, with a pivotal point, I was so aware of it, it's the best feeling of my life,” said the 23-year-old Rahm.
Fellow Spaniard Sergio Garcia became the leading points-scorer in Ryder Cup history with his 2 and 1 win over
American Rickie Fowler. The 38-year-old, a controversial wildcard pick by Bjørn after a poor season, passed Nick Faldo's previous mark of 25 points, reaching 25.5 since making his debut in 1999.
Rahm's triumph came after Denmark's Thorbjørn Olesen, in the seventh match to tee off, powered to an impressive 5 and 4 win over the in-form Jordan Spieth to quell the US team's momentum.
Jim Furyk's men had pulled to within a point after claiming 3.5 points from the first four matches on Sunday, with Justin Thomas defeating Rory McIlroy 1-up in the opening clash.
A calamitous 18th hole for McIlroy saw the Northern Irishman find the bunker with his tee shot, as well as the water, as Thomas picked up his fourth point of the competition.
Paul Casey and Brooks Koepka halved the second match, while Webb Simpson beat Justin Rose 3 and 2 after carding six birdies to put away the Olympic champion on the Albatros course.
Tony Finau romped to a 6 and 4 win over fellow rookie Tommy Fleetwood, denying the Englishman the chance to become the first European to win five points from a possible five at the Ryder Cup.