McIlroy headed into the last-16 clash after losing just three holes during the group stage but found himself three down after 10 and was set for an early departure.
But successive birdies at the 12th and 13th helped McIlroy cut the gap to just one before a messy 16th hole from the Northern Irishman ended his chances of progressing.
Woods sealed the win with a fine putt from 13 feet at the 17th to set up a last-eight match with Lucas Bjerregaard.
It was the first match-play contest between golf’s most marketable players, who have combined for 18 major championship titles and 778 weeks at world No.1 during their careers, and they did not disappoint fans at Austin Country Club.
The pair parred the opening hole before McIlroy scrambled for another par at the second as Woods asserted the early pressure.
The 14-time major champion missed more birdie chances at the next two holes as McIlroy clung on, only for Woods to finally make the first move with a gain at the fifth from four feet.
McIlroy looked set to regain that hole at the sixth when Woods found the fairway bunker but the American fired a fine five wood just short of the green before almost holing out with a superb up-and-down.
The 29-year-old could not match it, pushing his birdie putt wide of the cup, with Woods sinking his effort to move two holes ahead.
Woods drained a tricky eight-foot putt to halve the eighth hole, while a missed birdie opportunity at the next by McIlroy ensured the 14-time major champion maintained a two-up lead at the turn.
Woods’ lead improved to three holes at the 10th when McIlroy missed a three-foot putt for a half before the pair shared the next hole.
McIlroy hit back at the 12th after reaching the fringe of the green in two at the par-five, rolling his 40-foot eagle putt to tap-in range to register his first birdie of the match and cut the gap to two holes.
There was no Woods magic at the 13th following Friday’s eagle, with the world No.13 firing his approach over the back of the green to give McIlroy a chance to close the gap to one, which he duly took.
Woods then scrambled his way to pars at the 14th and 15th before McIlory appeared to continue his momentum with a 395-yard drive at the 16th, with the American in the bunker.
However, after negotiating his way back on to the fairway, the American found the green, while McIlroy could only dump his second shot into the rough – narrowly avoiding the bunker. The world No.4 then hit his third shot out of bounds to see his challenge fade before his fifth shot from the same spot found the greenside bunker just yards in front of him.
There was brief hope for McIlroy when Woods’ tee-shot at the 17th pulled up short but after the Northern Irishman missed a birdie attempt, the American rolled in the match-winning par putt to the roars of the Austin crowd.