McDowell stuck his approach to 4-feet on the 18th hole to set up a birdie for a 2-under 68 on Saturday. Furyk followed moments later with a 15-foot putt on the 17th to regain a share of the lead, and closed with a par for a 70.
At 1 under, the two major champions are the only players at the unforgiving Olympic Club course under par.
Fredrik Jacobson shot a 68 to move two strokes back.
Woods wasted his share of the 36-hole lead with six bogeys and only one birdie for a 75. The 14-time major champion will begin the final round five shots back.