The Americans led in all six fourball matches prior to a rain delay midway through the afternoon's action, but eventually had to settle for a 3 1/2 to 2 1/2 advantage over the International Team.
Brandon de Jonge, who made eight birdies Thursday at the Muirfield Village Golf Club, missed a 20-foot birdie putt on the final green that would have left the teams tied after the opening day.
Woods and Matt Kuchar teamed for one of the lopsided American victories, defeating Angel Cabrera and Marc Leishman 5-and-4. Johnson and Jason Dufner combined for a 5-and-3 decision over Branden Grace and Richard Sterne.
The International Team's comeback was started by Jason Day and Graham DeLaet, who fell behind by three holes before the rains brought a suspension of play. They rallied to defeat Hunter Mahan and Brandt Snedeker 1-up when Day made a 20-foot birdie effort at the 18th.
Hideki Matsuyama then hit his second shot at the 18th to within a foot for a birdie that lifted the International Team to a halve in his match. Matsuyama and Adam Scott paired to finish even with Bill Haas and Webb Simpson.
Louis Oosthuizen and Charl Schwartzel also came from behind to defeat Phil Mickelson and Keegan Bradley 2-and-1 while Steve Stricker and Jordan Spieth brought the United States a 1-up victory over Ernie Els and de Jonge.
Six foursomes matches will be played on Friday with five fourball and five foursomes matches set for Saturday. The 12 singles matches played on Sunday will conclude the competition.
The United States holds a 7-1-1 record in the event.