Johnson birdied the next two holes, added another on the 15th and another on the 17th and finished with a 5-under-par 66.
Despite the putting problems, the 14-times major champion shot a respectable one-under-par 70 to trail leader Dustin Johnson by 10 strokes heading into Sunday's final round.
Woods, a seven-time victor of the tournament but playing it for the first time since it moved to Mexico, made an early run, but a four-putt double-bogey at 15 followed by a three-putt bogey at 16 saw him post a one-under 70 that left him 10 adrift on 207.
Tommy Fleetwood, who began his second round with back-to-back eagle twos, shot 65 to lie tied fourth with Sergio Garcia, with Ian Poulter tied sixth after two 68s.
Woods had a 5-iron from the middle of the fairway on the par-5 15th, put it into a bunker, blasted out weakly to 25 feet and four-putted for a double bogey, with the last three putts from just outside 3 feet. He hit trees with three consecutive shots and had to two-putt from across the green to salvage a double bogey.
Johnson now leads the tournament after two days on 11-under, while McIlroy isn't far behind on nine-under.
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"Especially after the eighth hole, playing the last 10 holes in four-under par to keep myself within some sort of touch of DJ is nice".
To make matters worse, Woods three-putted the very next hole for bogey, and dropped back to 5-under for the tournament.
Rory McIlroy remains second after a 68, mixing seven birdies with four bogeys. It shot straight up in the air and hit the tree in front of me.
McIlroy said spotting Johnson four shots is a "tall order" to overcome. "He's going to be very tough to beat", McIlroy said.
He did that save for the 10th hole.