Spieth’s bogey-free 65 elevated the defending champion to nine under par alongside his housemate Kevin Kisner, with Xander Schauffele completing the three-way tie at the top after a day of great scoring in friendly conditions at the historic Angus links.
Woods will go into the final round just four off the pace after he posted his best score in an Open since he last lifted the Claret Jug at Hoylake in 2006, while Rory McIlroy’s hopes were hit by two bogeys over the last three holes as he also closed on five under.
English pair Justin Rose and Chris Wood had earlier set the tone for the day and given encouragement to the later starters as the leaderboard featured a sea of red numbers against a course unprotected by the elements.
Rose matched the best Open score at Carnoustie with his seven-under 64 which gave him the early clubhouse lead on four under, while Wood was seven under with five to play until his chances of a record-breaking round were scuppered by a bogey at the long 14th.
Both completed their rounds long before the overnight leaders teed off, while Woods soon set the pulses racing when he began to make significant strides in the right direction, with birdies at the fourth and sixth followed by three in a row around the turn.
The 14-time major champion added another at the 14th to claim a share of the lead at the time, although he suffered his only setback of the day at the 16th before a battling par at the last, where his tee shot came perilously close to ending up in the Barry Burn, completed an excellent 66.
But Woods would end the day four behind the top trio, with Spieth beginning his 65 in scintillating style when he drove the green and nailed the 10-foot putt for the only eagle of the tournament at the opening hole.
Spieth plotted his way meticulously around Carnoustie while picking up another shot at the sixth, and he added three more on the inward half to put himself in prime position to become the first man to successfully defend the title since Padraig Harrington in 2008.
Schauffele upstaged playing partner McIlroy with an impressive 67, recovering from an early bogey with three consecutive birdies from the fifth, and he made three more over the last five holes while dropping one shot at 17.
Kisner, who has held at least a share of the lead since opening with a 65, produced another composed performance in just his fourth Open appearance, collecting three birdies with no blemishes in his 68.
Kevin Chappell birdied the last to cap a 67 which got him to seven under and one ahead of the in-form Italian Francesco Molinari, whose poor finish to his Friday 72 was a distant memory by the time he had signed for a six-birdie 65 and claimed outright fifth.
Molinari will accompany Woods over the final 18 holes, while McIlroy will have an earlier tee time than he would have liked after an erratic 70 featuring five birdies offset by four costly bogeys.
After dropping a shot at the fourth, McIlroy revived his fortunes with consecutive birdies at six and seven before another at the 11th pulled him within touching distance of the leaders, although he then missed the green at the 12th and bogeyed for the second day running.
McIlroy stormed back to seven under with bounce-back birdies at 14 and 15, but he undid his good work when he three-putted from off the green at the 16th before a scrappy final hole resulted in another dropped shot.
Overnight joint-leader Zach Johnson lost ground on Spieth and co when he followed a bogey at 11 with a six at the next as he stumbled to a 72 which left him on five under along with Woods, McIlroy, Alex Noren, and last year’s runner-up Matt Kuchar.
The seven-way tie for sixth was completed by Tommy Fleetwood, who was just one off the lead overnight but slipped to three under when he dropped three shots in two holes at 12 and 13. But the Englishman battled back into contention with three birdies in four holes around another bogey at the 16th as he salvaged a level-par 71.