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, and Rahm held his nerve to beat Woods and leave the 14-time major winner without a single point.
Woods went into the week on the back of a first tournament 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. And only one overall triumph, in 1999.
Ian Poulter took his side to the brink of victory by beating world No 1 Dustin Johnson at the last and fittingly it was Molinari who had the honour of officially securing the win, Phil Mickelson conceding after dumping his tee shot into the water on the 16th to suffer a record 22nd defeat.
Molinari had partnered Fleetwood to four pairings victories and then became the first European player to compile a perfect 5-0 record in the contest. American Larry Nelson is the only other player to have done so, back 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. [Yesterday] it was hard to get the energy, but I stepped it up when I had to.”
Two up with three to play, Rahm appeared to hand 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 that 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 of his players.
“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 Henrik Stenson made it three wins out of three by beating Bubba Watson and fellow wild card Sergio Garcia defeated Rickie Fowler to become the top points scorer in Ryder Cup history.
Garcia thanked captain Bjorn for showing faith in him after delivering three record-breaking points to help Europe regain the Ryder Cup.
The Spaniard surpassed Sir Nick Faldo to become the most successful points-scorer in the history of the competition with 25.5.
Garcia followed up victories in the foursomes and fourballs over the first two days to beat Fowler 2&1.
The 38-year-old had been a controversial pick after missing the cut in all four majors this year.
Garcia said: “I don’t usually cry, but I couldn’t help it. What a week. It’s been a rough year, but we fought hard.
“Obviously I’m so thankful for Thomas for picking me and believing in me and I’m so happy to get the cup back here in Paris and in Europe.”
It was Garcia’s ninth appearance in the Ryder Cup and he was pleased to set a new individual record, although he stressed victory mattered far more.
He said: “It means a lot, but, at the end of the day, I’ve always said it’s about the team and I’m happy that I was able to help. I’m happy that I was able to help with some of the points.
“It’s a bonus. It’s something that I never thought I would have the possibility of doing, and now that it’s happened I’m just so happy.”
Bjorn hailed the impact of Garcia on the course and in the team room.
The Dane said: “I think Sergio’s Ryder Cup story tells itself – it’s a whole thing of its own, and it’s a brilliant story. It’s a fantastic achievement.
“He is very much the centre of what we do. He walks into the room, he is right in the middle of everything. Sergio’s done amazing.”
Bjorn will pay a personal price for his side’s emphatic Ryder Cup victory aftre agreeing to get a tattoo of the scoreline if Europe regained the trophy at Le Golf National,.
Poulter let the cat out of the bag in the team’s press conference, saying: “I don’t know when it’s going to get done, but we had some extra motivation this week to make sure we put our hands back on that trophy.”
Asked where the tattoo would be, Bjorn said: “Let me put it this way, it’s going to go on a part that only Grace [his partner] will see.
“I might have to send them a picture, well, then you’ll all see it as well, I guess.” He added: “The worst decision I made all week.”