Woods shot five-under 67, his lowest round at Augusta since 2011, to move to 11-under for the tournament and leave himself just two shots behind British Open champion Francesco Molinari.
At six-under-par 138, the American is just one shot off the leaders heading into "moving day".
But while a 43-year-old Woods was doing battle for a fifth green jacket at the top of a star-studded leaderboard, McIlroy was putting the finishing touches to a second consecutive 71 to finish one under par.
The tournament record is 63. He's at 13 under par and two strokes ahead of Woods.
Ryder Cup team-mate Tyrrell Hatton reacts to Francesco Molinari's unbelievable record of just one bogey in 54 holes with a video of a shocked audience member from Australia's Got Talent and three applause emojis.
Woods finished tied for sixth at The Open, where Molinari won, and was second at the PGA Championship, where Koepka won his third major. It's the first time three players have shot 64 or better in a single round of the Masters. Finau now leads the tournament at 9-under and Simpson, Xander Schauffele, Adam Scott and Brooks Koepka are one shot back.
But his biggest piece of luck came at the par-five 13th, where he hooked his drive so far left that the ball seemed more likely to end up in adjacent Augusta Country Club than stay in Augusta National.
Numerous times throughout his career, Woods has opted to go with a strategy of laying back off the tee and trusting his mid or long irons to pick up the slack.
Italian Molinari carded an error-free 67 for his best round in eight Masters that put the British Open champion in contention for a Green Jacket to go along with the Claret Jug he hoisted previous year. It added up to a 68 for the Australian who won the Masters six years ago, and has gone three years since his last victory. "That's where my advantage is, anyway, but in emphasis here at Augusta National".
Finau will be the wildcard in the last threesome.
"I've just been making too many mistakes". The leaders were to tee off at 9:20 a.m.
"All parts of my game I'm very confident in at the moment, and you know, when I am hitting it well, that's where I play my best golf", Scott said.
Woods, who entered the week an incredible 156 under on the par 5s for his career (he's a cumulative 68 over on the other holes), has had to spread his efforts around.
"He (Woods) obviously loves this place and he's playing great golf", Molinari said.
Because of the change in starting times, Finau gets to play alongside Woods - his golfing hero. "That really gave me confidence and I always said to myself looking at the score board that I was there or thereabouts without kicking in on the front nine".