Score after Saturday's matches: USA 5-11 Europe
European captain Bernhard Langer has rewarded England's David Howell with the dream slot as the man who could hole the winning putt in the Ryder Cup.
Howell won a vital point in Saturday's fourballs with fellow rookie Paul Casey, and goes off fourth against Jim Furyk in Sunday's 12 singles matches.
Europe lead 11-5 and need three points to retain the trophy but Howell has the first chance to seal outright victory.
Casey will lead off the European team at 1700 BST against Tiger Woods.
Langer opted to sprinkle his more experienced players throughout the singles draw, rather than load his top order as captain Sam Torrance did with great effect at The Belfry in 2002.
Spain's Sergio Garcia will take on Masters champion Phil Mickelson in the second match, while Northern Ireland's Darren Clarke is off third against Davis Love.
Ireland's Padraig Harrington, Europe's highest-ranked player at eight in the world, will tee off in the 11th match as an insurance policy against an American resurgence.
"Not in my wildest dreams would I have imagined this lead and I like what I
see in the draw," said Langer.
"I am not scared at all. I think they are going to
win five or six points and that would be fantastic.
"I would be very surprised if we don't get two points from the first three. We might not have the superstars we had in the 1980s but we have great players."
But Garcia, who has won 3½ out of four points so far, warned against European complacency.
He said: "If you go in with that mentality you are probably going to lose.
"We have to go out and play as hard as we can in Sunday's first few matches to give ourselves a good chance."
US captain Sutton admitted his method of selecting his order of play in the singles was less scientific - he put them out in the order they qualified.
"That was as good a way as any," said Sutton.
"The top three players are the strongest players we have got.
"We have a large deficit and maybe the Tiger Woods and Phil Mickelson you all
know have not been here this week. But there is a good chance they will be there
on Sunday and if they are what's going to happen?
"If they do show up they will be formidable opposition.
"All I'm going
to tell them is what they have already heard in their own mind. I'm not going to
whip them. They've been whipped already in the way they've played.
"I believe in my heart they can win."
The Americans fought back to win from a 10-6 deficit going into the singles at Brookline in 1999 and both Woods and Mickelson insisted a similar reverse was possible.
Woods said: "We're going to have to obviously get off to a great start and just hopefully it just snowballs and rolls.
Casey will kick-off Europe's quest for glory in the singles
"You never know. You never know what could happen. Obviously, we're going to be very optimistic about our chances, but we need to have guys just flat-out win matches, period."
Mickelson added: "It will take something exceptional and it's very unlikely but we won't give up.
"It will be very difficult but I wouldn't count us out just yet."
After a stunning first day which Europe took 6½-1½, the visitors stretched their lead to 8-4 after Saturday's fourballs.
And they won three more matches in the foursomes to forge their biggest-ever margin going into the singles.
The previous best was 10½-5½ in Dublin, Ohio in 1987 and at Valderrama in 1997 and both times Europe won.
Victory at Oakland Hills would be Europe's fourth success in the last five matches.
Ryder Cup singles draw: (all times BST)
Tiger Woods v Paul Casey (1700)
Phil Mickelson v Sergio Garcia (1711)
Davis Love v Darren Clarke (1722)
Jim Furyk v David Howell (1733)
Kenny Perry v Lee Westwood (1744)
David Toms v Colin Montgomerie (1755)
Chad Campbell v Luke Donald (1806)
Chris DiMarco v Miguel Angel Jimenez (1817)
Fred Funk v Thomas Levet (1828)
Chris Riley v Ian Poulter (1839)
Jay Haas v Padraig Harrington (1850)
Stewart Cink v Paul McGinley (1901)