Woods only beat the cut by one shot on Friday evening after posting a second-round 71. But he was back to his best on Saturday as he rose from tied 68th into the top ten by early afternoon, and he was even eight-under through 12 holes.
At one stage Woods looked capable of threatening his career-best round of 61, or going even lower, but the birdies then dried up.
The 14-time major winner, fit again after an injury-enforced spell off the tour, had struggled off the tee over the first two days and did not have too many realistic birdie chances.
But on day three the 42-year-old rolled in birdies on the opening two holes – the first one from 15 feet – and the putter stayed hot on the fourth and fifth greens.
Woods drained a 17-footer on the fifth and further birdies followed at seven and nine, where he two-putted from 40 feet.
It saw Woods make the turn in 30, two shots better than his previous best front-nine score at Sawgrass and his lowest nine-hole score since 2009.
Woods picked up another shot at the par-five 11th and sank an eight-foot putt for birdie on 12. But he made his first mistake at 14 when he caught an awful lie from a wild tee shot and an underwhelming chip resulted in a two-putt bogey.
Woods still had the course record of 63 in his sights as he began the tough last three-hole stretch.
But an eight-foot birdie chance at the 16th lipped out and his shot to the island green 17th hole ended 43 feet away, although he tidied up in par.
Another birdie chance slipped away at the last after he had hit a nine iron from 193 yards to within 17 feet.
Jordan Spieth was also among the early starters and, like Woods, sprinted up the leaderboard with a seven-under 65.
Spieth, who has struggled at Sawgrass in recent years, had nine birdies and two bogeys in outshining playing partner Justin Thomas, who carded a 68.
Woods and Spieth were both seven shots behind Webb Simpson, who was about to start his third round holding a five-shot lead over Charl Schwartzel, Patrick Cantlay and Danny Lee.
On the European Tour, Mike Lorenzo-Vera will take a two-shot lead into the final round of the Rocco Forte Sicilian Open after firing a brilliant 63 on day three at Verdura Golf Club.
The Frenchman has never won on the European Tour in over 150 appearances, but he will have an excellent chance of changing that today after taking advantage of calmer conditions on the Sicily coast to get to 15 under.
Playing partner Joakim Lagergren also fired a 63 to sit at 13 under alongside Julien Guerrier, one shot clear of Lucas Bjerregaard.
“If I bring the good game, I can win for sure,” Lorenzo-Vera told europeantour.com. “If someone just doesn’t go crazy and shoot 10 under if it’s good weather, I’ll have a really good chance.
“I think it was better today. Everything came together around the turn so it was cool. It was easier for sure, but in that case you have to be careful because you can lose your focus and you know someone is going to shoot low, so you just have to play the game and try to keep calm, wait for the birdies to come.”
Frenchman Guerrier entered the day in a share of the lead, but Lagergren was the first man to get past him, adding to gains on the second and third with five in a row from the sixth to lead by two.
An eagle on the ninth had Guerrier back level with the Swede, but, while they could only pick up one further shot, Lorenzo-Vera made a charge through the field.
He turned in 33 with birdies on the second, seventh and ninth, adding further gains on the 11th, 12th, 13th, 15th and 16th.
England’s Andy Sullivan was six shots off the lead after a 65.