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.
But seven shots back from Johnson's 16-under par it would take something special to overhaul the former world No. 1.
McIlroy had a slow start - he was six shots off Johnson's lead after eight holes - but played the final 10 holes in 4 under.
A stunning pitch from a tough lie from Johnson set up a tap-in birdie at the 15th, where McIlroy hit a brilliant approach under the trees but missed his 12-foot eagle attempt. These green here are pretty tricky and I keep giving myself opportunities to make birdies.
Dustin Johnson's stellar play all weekend in Mexico meant that Tiger Woods was out of contention for a win midway through his third round on Saturday - but Woods' solid effort was spoiled by an ugly double-bogey 7 on a par-5.
"I feel good just about everywhere", Johnson told Golf Channel, when asked about the state of his game.
Johnson built a 16-under par total of 197 at Club de Golf Chapultepec, where McIlroy fired a three-under 68 for 201. He was too firm with his bogey putt and lipped out again, and finally tapped in for double.
Woods birdied the opening two holes, tapping in at the first and then pitching to two feet from 137 yards, but he double bogeyed the 15th after taking three putts from three feet, before finishing with his sixth birdie of the day. He's among the four players tied for ninth at 207, 10 shots off Johnson's lead.
The following three-putt was the first time ever in his PGA Tour career in which he as followed up a four-putt with a three-putt.