Jose Maria Olazabal's side need to replicate the biggest comeback in cup history to retain the trophy, but both Olazabal and Lawrie know from bitter personal experience that it is possible.
Both men were in the European side which led by the same score at Brookline in 1999, only to lose the first six singles matches and the overall contest by a single point, with a superb US comeback marred by the infamous invasion of the 17th green during Olazabal's clash with Justin Leonard.
"There's still 12 singles matches tomorrow [Sunday]," said Lawrie, whose win over Jeff Maggert in the final match that day was one of only three for the European side. "We can all win tomorrow [Sunday] and why not?
"It's certainly not over. It would be nice to make a big comeback tomorrow [Sunday], so we'll see what happens."
It took two remarkable last-hole wins by Luke Donald and Sergio Garcia and Ian Poulter and Rory McIlroy to trail by just four points, but Lawrie added: "It's always possible. Until it's impossible to do it, then you fight on, certainly for Jose Maria this week.
"No one is going to be giving up."
Lawrie, making his first appearance in the contest since Brookline and the year of his Open Championship win, suffered his second fourballs defeat on Saturday, partnering Nicolas Colsaerts to a one-hole loss to Dustin Johnson and Matt Kuchar.
The Scot managed just one birdie in 18 holes but made a crucial par on the 16th to get back to all square before Johnson holed from 22ft for a decisive birdie two on the 17th.
"It's all about putting," the 43-year-old added. "Ryder Cup always has been. All the players are top players, world-class players. We can all play, it's just a matter of who holes the putts and today [Saturday] the American boys holed the putts at the right time."