Jose Maria Olazabal's men trail the Americans 10-6, but if more of them - preferably all of them - play like Ian Poulter did again on day two it might just happen.
Poulter produced a spectacular five closing birdies, the last of them from 10 feet in near darkness, to give himself and world number one Rory McIlroy victory on the final green over Jason Dufner and Zach Johnson.
The Englishman, winner of all his three games so far, now has a phenomenal record of 11 wins and only three defeats in his cup career.
It followed another last green win for Luke Donald and Sergio Garcia over Steve Stricker and Tiger Woods, who is now the joint holder of the record for most defeats by an American in the event's history.
Woods suffered his third defeat of the week after being left out of the morning foursomes - something that had never before happened to him in his Ryder or Presidents Cup career.
Poulter said: "We needed to get something going. We had to make birdies and wow - five in a row. It was awesome.
"I had the world number one backing me up and it allowed me to hit some shots."
They had been two down with six to play and McIlroy added: "At least I had a little bit to do with it, making birdie at 13, but when Ian gets that look in his eyes it's really impressive.
"It was incredible making that run like we did."
The only side to win from four points down going into the 12 singles was the American one at Brookline in 1999 - the match remembered for Justin Leonard's 45-foot putt against Olazabal on the 17th green.
It sparked controversial premature celebrations because Olazabal still had a putt to keep the game and overall match alive - and this year's American captain Davis Love was among those who came on to congratulate Leonard.
Since the competition began in 1927, however, the United States have never lost more than a two-point advantage on the last day.
Love said of losing the last two games of the day: "We're not disappointed - our guys all played real well.
"We lipped out a bunch of putts. We've not lost a segment (session) yet."
They took the morning foursomes 3-1 to make it 8-4 and the fourballs were shared 2-2.
Olazabal stated: "Those last two matches were massive. It gives us a chance - it's been done in the past.
"Things have not gone our way, especially on the greens, but I believe momentum will come our way. Why not tomorrow [Sunday]?"
The gap was six points when Masters champion Bubba Watson and US Open winner Webb Simpson hammered Justin Rose and Francesco Molinari 5&4 and then Dustin Johnson and Matt Kuchar edged Nicolas Colsaerts and Paul Lawrie on the last.
Lawrie won the 16th to square it after Europe had been two down early on, but Johnson sank a putt of over 20 feet at the next and the 18th was halved in par.
"It's probably the loudest roar I've ever heard, especially in this situation," Johnson said.
"My partner had been carrying me all day pretty much, but that was one of the best putts I've ever made."
Poulter and McIlroy were two down with six to play, but McIlroy's 15-footer at the 13th set up a thrilling climax.
Poulter got up and down from bunkers at the next two, first for a half and then for a win to level the match, rolled in a 15-footer to take the 16th and, with Zach Johnson close, made no mistake from nine feet for a half in birdie twos at the short 17th.
Dufner hit the best approach of the four into the last, just three feet from the flag, and holing it - he had the chance to go first because of where his partner was - Poulter had to make his from more than three times as far.
That he did adds another chapter to an amazing cup story for one of Olazabal's two wild cards.
He won his last four games in the 2008 defeat, three out of four in the 2010 win and now his first three this week.
Leaving him out of the first day fourballs is perhaps the one thing Olazabal regrets doing, but if Europe do pull it off and he wins again it will look a masterstroke.