McIlroy began the day five shots off the lead held by Ryder Cup rival Patrick Reed, but birdied the third and fourth and then nearly holed his tee shot on the par-three sixth. Given his position seven shots back of the lead, he won't be able to afford such mistakes Sunday.
20 Masters champions: Sergio Garcia (2017); Danny Willett (2016); Jordan Spieth (2015); Bubba Watson (2012, 2014); Adam Scott (2013); Charl Schwartzel (2011); Phil Mickelson (2004, 2006, 2010); Angel Cabrera (2009); Trevor Immelman (2008); Zach Johnson (2007); Tiger Woods (1997, 2001, 2002, 2005); Mike Weir (2003); Vijay Singh (2000); Mark O'Meara (1998); Jose Maria Olazabal (1994); Bernhard Langer (1993); Fred Couples (1992); Ian Woosnam (1991); Sandy Lyle (1988); and Larry Mize (1987).
Reed led by two shots entering the third round.
"It's going to be awesome to come out tomorrow and have a lot of Augusta State guys out here cheering me on", Reed said - although he noted that the stately confines of Augusta National would make for a far different experience than a rowdy Ryder Cup.
Still fighting his way back to top form after undergoing spinal fusion surgery last April, Woods offered glimpses of the skill and talent that lifted him to 14 major titles but was never in serious contention. But at the same time, he's trying to go for the career Grand Slam. "I knew it was going to be a dogfight".
No matter how he finishes, he will leave the course with two huge highlights on "Moving Day".
McIlroy had already made three birdies before a chip-in eagle at the eighth and then scrambled to save par after his ball found the azaleas behind the 13th green.
Even removed by one pairing Saturday, the two put on a good show with full awareness of what the other was doing.
Five predictions for the 2018 Gulf Air Bahrain Grand Prix
Australia's Daniel Ricciardo's Red Bull will start the race fourth on the grid. "The Ferrari was quicker today", said Bottas. Sebastian Vettel is perfectly placed to strike another early blow to Lewis Hamilton's Formula One title defense.
Woods made his fourth of the day when he sank a 4-foot putt at the par-5 13th. He sank the comeback effort and pumped his fist in celebration. "And when he tried to make a counter, I was able to always stay ahead and keep going".
It's the fifth time in 46 holes that Reed strung together back-to-back-to-back birds.
Victory is well out of sight for the 14-time major champion, but Woods hopes to finish with a flourish on Sunday.
He bogeyed the par-three 12th after sending his tee shot into Rae's Creek and while he did well to card a pair of birdies over the next three holes, he dropped another shot at the par-three 16th.
Three-time Masters victor Phil Mickelson made the cut on the number at 5-over 149. He is at 7-over 223 heading into the final 18 holes Sunday.
Reed took charge of the third round at Augusta National with three birdies on the hop from the eighth before eagling both 13 and 15 en-route to a 67.
Third-round leader/co-leader has won Masters 44 (out of 81) times.
"You can put it either way", Reed said, adding that he "felt fine" after sleeping on the halfway lead. He made contact the second time, missed the green and eventually suffered a triple-bogey 7 en route to shooting 2-over 74 to be at 7 over.
The only Masters champion who was not inside the top 10 after 54 holes was Art Wall Jr.in 1959 (T-13th).
Spieth flirted with breaking Augusta's long standing course record (63) as he logged nine birdies through the first 16 holes, but finished with an electrifying eight-under 64, as Fowler broke out of a first nine funk to post four birdies on the back nine, including a clutch putt on the last.