The home side’s 10-6 overnight lead meant they needed just four and a half points from the 12 singles for a ninth win in the 12 contests staged since the last American win on European soil in 1993.
Nerves were jangling at Le Golf National when the gap was reduced to just a single point as Justin Thomas beat Rory McIlroy on the 18th, Webb Simpson defeated Justin Rose and Tony Finau thrashed the previously unbeaten Tommy Fleetwood.
However, Thorbjorn Olesen responded to being left out on Saturday by fellow Dane and Europe captain Thomas Bjorn with a resounding victory over three-time major winner Jordan Spieth 5&4, while Rahm held his nerve to beat Woods and leave the 14-time major winner without a single point.
Woods came into the week on the back of a first win in five years and supposedly more invested than ever in the contest having acted as a vice-captain during the 2016 victory at Hazeltine.
However, the 42-year-old lost all three pairs matches - making it eight defeats in a row stretching back to 2010 - and the singles loss to Rahm means an overall record of 13 wins, 21 defeats and three halved matches in eight appearances as a player.
Ian Poulter took his side to the brink of victory by beating world number one Dustin Johnson on the 18th and fittingly it was Molinari who had the honour of officially securing the win, Mickelson conceding after dumping his tee shot into the water on the 16th to suffer a record 22nd defeat.
After four wins alongside Fleetwood, Molinari is the first European player to compile a perfect 5-0 record in the contest, with American Larry Nelson the only other to do so in 1979.
“It means so much, more than majors, more than anything. It’s hard not to get emotional,” the Open champion said. “They had probably the strongest team ever and we were just so good, it is unreal. Today it was hard to get the energy but I stepped it up when I had to.”
Two up with three to play, Rahm handed Woods a lifeline by three-putting the 16th, but then smashed a wonderful 356-yard drive down the 17th and hit his approach to three feet to set up a winning birdie.
“There is a lot going on for me right now,” Rahm said. “The memory of Seve (Ballesteros) and my grandpa, who would have loved to be here, who passed away at the (US) PGA.
“To beat Tiger, one of the all-time greats, if not the greatest, on the Sunday, at a pivotal point, it is the best feeling of my life.”
After losing the first three matches on Friday morning, Europe won 10 of the next 11 against an American team which was the strongest assembled since the world rankings were introduced with an average ranking of 11.2
“They have been amazing, the 12 of them, unbelievable”, Bjorn said. “They were determined to set out and do the job themselves and for me it was an easy job to guide them in that direction.
“It was getting tight a little bit, but I had belief and faith in this team that they were going to deliver and they delivered more than they should have.”
With the contest over there was still time for Henrik Stenson to make it three wins out of three by beating Bubba Watson and fellow wild card Sergio Garcia to make a mockery of questions about his selection by defeating Rickie Fowler to become the top points scorer in Ryder Cup history.
No wonder then that United States captain Jim Furyk admitted his team had lost to the better side.
“At the end of the day you have to tip your cap, they outplayed us this week and Thomas was the better captain,” Furyk said.