Woods rallied to defeat fellow American Patrick Cantlay 4&2 and Northern Irishman McIlroy prevailed over England's Matthew Fitzpatrick by the same score.
That put both in Saturday's round of 16.
"I'll be ready," Woods said. "We’ve played tournaments together, battled each other down the stretch at events, but never in a match play situation, so this will be fun.”
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The two were paired in the final group of last year's Tour Championship in Atlanta, which Woods won while McIlroy shot a 4-over 74 to finish tied for seventh.
“I just need to put up a better fight than I did," McIlroy said. "I didn't have much of a game that day."
Woods, the 13th seed, needed a win and a loss by Brandt Snedeker to avoid elimination. He got both.
Woods posted four birdies and an eagle in a six-hole stretch on the back nine at Austin Country Club to claim the victory while Snedeker was beaten 6&4 by Aaron Wise. Cantlay had been 2 up at one point before Woods' rally.
"I made a big putt at 11 and got it rolling from there," the former world number one said of a 21-foot putt that pulled him even with Cantlay.
Woods then moved out front with a six-footer at the 12th and got the crowd excited with an eagle from 82 yards.
"I just tried to get up there close, and the next thing you know the ball falls in," he said. Woods followed with birdies at 14 and 16. Fourth-seeded McIlroy never trailed in his match with Fitzpatrick as he remained undefeated for the tournament.
"I played well," said the Northern Irishman. "I think I made six birdies in 16 holes so I am right where I need to be."
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