Europe lead the United States 10-6 heading into Sunday's Ryder Cup singles after a one-sided penultimate session at Gleneagles in Scotland.
Europe won Saturday afternoon's foursomes 3½-½ to move to within four points of retaining the trophy.
Europe lost Saturday morning's fourballs 2½-1½ before making it 7-1 in foursomes matches over two days.
However, the US led by the same margin after two days at Medinah in 2012, only for Europe to clinch a remarkable win.
The US also came back from 10-6 down at Brookline in 1999.
"We are not finished. We have a lot of work to do and we have to be on it in the morning," said Europe captain McGinley, who was a vice-captain in Chicago two years ago.
US captain Tom Watson, who left Phil Mickelson out for the entire day for the first time in 10 Ryder Cups, said: "We have come back from 10-6 before. They know it and I'll reiterate it.
"Our rookies played some magnificent golf and we have the players to come back. Credit to the Europeans, they played some great golf. It seems the foursomes is their forte."
Having trailed 5-3 after day one, the US reduced the deficit to one point after the morning fourballs.
But Lee Westwood and Jamie Donaldson beat Zach Johnson and Matt Kuchar 2&1 before Victor Dubuisson and Graeme McDowell thumped Rickie Fowler and Jimmy Walker 5&4 in the last match out.
Rory McIlroy and Sergio Garcia saw off Jim Furyk and Hunter Mahan 3&2, before Justin Rose holed a pressure putt at the last to ensure he and Martin Kaymer halved with Jordan Spieth and Patrick Reed.
Englishman Westwood is now the joint-fifth most successful player in Ryder Cup history.
Westwood, 41, has scored 23 points in nine Ryder Cups and leapfrogged the late Seve Ballesteros to move level with American legend Arnold Palmer.
Another point in Sunday's singles would put him above Colin Montgomerie and Billy Casper on 23½ and joint-second with Germany's Bernhard Langer, with only Sir Nick Faldo ahead of them on 25 points.