After red-hot duo Phil Mickelson and Keegan Bradley inspired the Americans to win the morning's foursomes session 3-1, Bubba Watson and Webb Simpson led the way in the afternoon's fourballs which were split 2-2.
Europe had trailed in three of the fourballs but Sergio Garcia and Luke Donald held off a fightback by Woods and Steve Stricker to win one up before Rory McIlroy and Ian Poulter came from two down to beat Jason Dufner and Zach Johnson by the same margin.
Matchplay specialist Poulter finished with five consecutive birdies on his ball, the last of them a 12-foot putt in fading light to give the Europeans a slim hope of retaining the Cup heading into Sunday's concluding singles matches.
A strong European team had arrived at Medinah closely matched with the U.S. but with several of their top players not at their best, they face a daunting challenge on the last day.
The Americans famously pulled off the largest Ryder Cup comeback at Brookline in 1999 when they trailed 10-6 going into the final day and the Europeans would have to match that to keep the trophy.
"Our boys are not making the putts," Europe captain Jose Maria Olazabal, who played in that year at Brookline, said midway through the fourball session. "Some of them haven't performed to their expectations."
Left-hander Watson and Cup rookie Simpson earned the U.S. their first point on a sunny afternoon when they beat England's Justin Rose and Italian Francesco Molinari 5&4 in front of highly energised crowds.
Watson sealed the win with a two-putt birdie at the par-five 14th, but U.S. Open champion Simpson paved the way to victory with a red-hot putting display as he piled up seven birdies on his own ball.
"Webb just was a little anxious, wanted to make all the putts," the long-hitting Watson said. "I did the par-fives and he did the rest, and that's pretty much all we did."
Dustin Johnson and Matt Kuchar added another point for the Americans when they held off a late charge by Belgian Cup rookie Nicolas Colsaerts and Scot Paul Lawrie to win one up.
With the match all square after 16 holes, Johnson effectively delivered the dagger blow when he sank a curling 20-footer to birdie the par-three 17th and trigger thunderous cheers from the crowd.
"It was probably the loudest roar I've ever heard," said Johnson, who had celebrated the putt with a sweeping fist pump. "In that situation, it was probably one of the best putts I've ever made."
Woods and Stricker, benched for the morning's foursomes after losing their first two matches, seemed down and out when four down to Spaniard Garcia and Englishman Donald.
However Woods, who had struggled badly with his game on the front nine, sparked after the turn with a run of five birdies in eight holes to take the match to the 18th.
Stricker, one of the best putters in the game, lined up a seven-foot birdie putt at the last that would have tied the match but his attempt lipped out.