McIlroy carded a 75 for a six-over total of 150 while playing partner Woods was given a belated two-shot penalty for an incident at the fifth hole and seems sure to miss out as well after having to sign for a 75, rather than a 73, for a tally of 147.
Woods now faces an anxious wait before the two-over-par projected cut of 146 is confirmed.
The American hit his ball into a desert bush at the fifth and, after consulting with the third member of the group, Martin Kaymer of Germany, the pair agreed he should be allowed a free drop because it had become imbedded.
But the regulations only allow for a free drop in such circumstances if the ball has landed on grass. Because it finished in sand, Woods was handed a retrospective two-shot penalty by officials as soon as his round ended.
"It's tough," Woods told Sky Sports television. "I didn't get off to a very good start today but I fought and got it back.
"I was right there and I felt if I could post even-par today I'd be right there for the weekend but evidently that was not enough."
Woods dropped shots at three of the first four holes before the incident at the fifth meant he had to sign for a triple-bogey seven instead of a bogey five.
The 14-times major champion's putter caught fire on the homeward stretch as he picked up strokes at the eighth, 10th, 14th, 15th and 16th but his late birdie rush could now prove in vain.