Reigning Masters champion Woods, a 15-time major winner, fired a two-over par 73 to stand on one-under 212 after 54 holes at TPC Boston.
That was only good enough for 67th place, one spot ahead of playing partner McIlroy, who fired a 74 to stand on 213 in third-to-last.
McIlroy endured triple-bogeys at the par-5 second and par-4 sixth holes, sandwiching his misery at the second between birdies.
Four-time major winner McIlroy missed the fairway, found the native area, hit the ball into the water off a rock and three-putted to make an 8 on the second, then needed four shots to find the fairway at six.
McIlroy opened and closed the back nine with birdies with a bogey at 14.
Woods birdied the second but followed with bogeys at three and five. He had seven-foot birdie putts to close the front and back nines but had three bogeys in four holes from the 11th to 14th.
Woods indicated he actually added a milestone moment to his storied career in an otherwise unspectacular round: when he tapped in for par at the 17th and told McIlroy it was his first time "ever" to putt with the flagstick in.
Until the Rules of Golf revisions went into effect last year, doing so would bring a penalty, and tour pros are only gradually adopting the practice.
"That's my first time," PGA Tour Live microphones caught Woods telling McIlroy.
"Ever?" McIlroy responded.
"Ever," Woods said before his caddie, Joe LaCava, sparked laughter when he added: "If he was three-under, no chance, Rory."
With COVID-19 restrictions discouraging use of the clubhouse, the superstar duo capped their round just as many weekend golfers do.
They were spotted having hamburgers at a picnic table not far from the media tent.