Fleetwood and Molinari defeated Tiger Woods, who played alongside Patrick Reed and then Bryson DeChambeau, in both the fourballs and foursomes to become the first European pairing to claim four points from their opening four matches.
They joined the American duo Larry Nelson and Lanny Wadkins who did it at The Greenbrier in 1979.
The tournament has also provided 48 hours of redemption for the Spanish golfer Sergio Garcia.
In the six months before the competition, he struggled with his game by missing the final stages of eight of his 11 tournaments. In defending the Masters crown last April just weeks after the birth of his daughter, Azalea, he needed 13 shots to sink the ball on the 15th hole at Augusta National. The fiasco tied a Masters record for most strokes on any hole.
But despite the disasters, Europe captain, Thomas Bjorn, selected him. And the 38-year-old has not disappointed.
He sunk a 20-foot birdie putt at the 17th hole during a fourball victory alongside Rory McIlroy on Saturday over Brooks Koepka and Tony Finau.
"I think that I've proved myself over and over," Garcia said at the end of the second day. "The only thing I can do out there is when I get called upon is just do my best, do what I do, do what I've always done here at Ryder Cups, and that's everything."
Saturday’s victory and a Friday foursomes triumph alongside Sweden's Alex Noren has furnished Garcia with a chance to break the all-time Ryder Cup points record.
Garcia, already the youngest player in Ryder Cup history from his 1999 debut at age 19, has won 24 and a half points, just off Nick Faldo's record of 25.
"When things don't go exactly as you plan or as you want it, and you keep missing cuts by one, it feels like it's kind of getting a little farther away and you want it to come back," Garcia added. “I talked to Thomas and I told him that I want to be a part of the team. I'm definitely glad to be here."
Jim Furyk, the US team captain, said he would need to inspire his players for the final day. Furyk highlighted the Brookline team of 1999 who rallied from 10-6 down to claim the prize.