Furyk teamed with Nick Watney for a 1-up win over the International Team of Adam Scott and Ernie Els and Mahan and Bill Haas beat Aaron Baddeley and Jason Day 2-and-1 in pushing the Americans to a 13-9 lead after Saturday's fourball matches.
The United States needs to collect four points over Sunday's 12 singles matches to retain the Presidents Cup, which it has held since 2000. No team has come from behind in Sunday's singles to win the Presidents Cup.
The International team cut into the lead by winning three of the five fourball matches later Saturday. That included Kim Kyung-tae and Y.E. Yang combining to beat Woods and Johnson 1-up after Kim provided the difference with a 15th-hole birdie.
Retief Goosen and Charl Schwartzel defeated Webb Simpson and Bubba Watson 2-up and Choi and Geoff Ogilvy were 1-up winners over Steve Stricker and Matt Kuchar. But wins by Furyk-Watney and Mahan-Haas pushed the Americans' advantage back to four points.
Earlier Saturday, the teams of Watson-Simpson and Mickelson-Furyk improved to 3-0, while Mahan and David Toms enjoyed their second win in three attempts.
Watson and Simpson won three out of five holes to close out the match and defeated Robert Allenby and Ogilvy 3-and-2.
Mickelson and Furyk trailed by two holes through the 10th, but then warmed up on the greens to defeat Baddeley and Day 2-and-1. Mickelson ended the match by making a 50-foot putt on the 17th hole.
Mahan and Toms coasted to a 5-and-4 decision over Goosen and Schwartzel.
The only International Team win in the morning session came from Els and Ryo Ishikawa, who posted a 1-up victory over Haas and Kuchar. Els and Ishikawa had lost their first two matches this week to Simpson and Watson.