Needing two more birdies for the magical number with five holes to play, Woods narrowly missed a nine-footer at the 15th and a six-footer at the 17th on the way to a nine-under-par 61 at Firestone Country Club.
His sizzling score equalled the course record, which he had previously tied in the second round in 2000, and gave him a commanding seven-shot lead with a 13-under total of 127.
Woods, who has triumphed a record seven times in the elite World Golf Championships (WGC) event at Firestone, is bidding to win his fifth title on the PGA Tour this year as he builds momentum for next week's PGA Championship.
"I had a lot of control today from tee to green and obviously the way I putted," the American world number one told reporters after totalling only 22 putts. "I felt I was in total control of my game.
"Obviously things like that don't happen every day, and it's fun when it all comes together and I was able to take advantage of it, especially on a golf course like this."
Asked whether he was at all disappointed to fall short in his bid to fire a 59, Woods replied: "Absolutely not, nope. 61 is pretty good. I'm not bummed.
"Would it have been nice to shoot 59? Yeah, it would have been nice. I certainly had the opportunity. I just had (to get) two more (birdies) out of five holes. I had two good chances at 15 and 17 to do it.
"But the par putt at 18 was even bigger," Woods said, referring to the 26-footer he sank from the back fringe to keep his card bogey-free, prompting him to raise his right arm skyward with a pointing finger in celebration.
Defending champion Keegan Bradley and Englishman Chris Wood carded matching 68s on the challenging South Course to end the day tied for second, with Swede Henrik Stenson (70) and American Bill Haas (68) a further stroke back at five under.
However, the tournament would now appear to be in Woods' hands to win or lose as he heads into the weekend at one of his favourite venues with a seven-shot advantage.
(Reporting by Mark Lamport-Stokes; Editing by Gene Cherry)
