The International team last led the event, held this year at Royal Melbourne in Australia, at the end of day one in 2005 and have lost the past seven editions of the tournament.
US captain Woods and Justin Thomas beat Marc Leishman and Joaquin Niemann 4&3 to put the first point on the board, but it all went wrong for the Americans from there on.
Louis Oosthuizen and partner Abraham Ancer also won 4&3 over Dustin Johnson and Gary Woodland, while Adam Scott, playing in front of his home crowd with South Korean An Byeong-hun, beat Bryson DeChambeau and Tony Finau 2&1.
And Adam Hadwin and Im Sung-jae ended one-up against Xander Schauffele and Patrick Cantlay.
Els said: “This is an unbelievable start. We haven’t had a start like this for years.”
Woods, meanwhile, added: “Just because we lost the session doesn’t mean the Cup’s over. There’s a long way to go, a lot of points available.
“The guys will regroup and we’ll come out tomorrow ready to go.”