Woods, winless in the majors since the 2008 U.S. Open, hasn't been this close to the lead going into the final round at a major since he was two shots back at Muirfield five years ago.
Woods underwent spinal fusion surgery in April past year and was arrested on suspicion of driving under the influence the following month when he was found asleep at the wheel of his vehicle.
Woods has not played in the Open since 2015 due to injury and is without a victory since 2013, while the last of his 14 major titles to date came a decade ago in the US Open.
For a few hours at a sun-drenched Carnoustie on Thursday it felt like 2000 all over again, or 2005 or 2006 as Tiger Woods rolled back the years and captivated a heaving British Open gallery.
His previous best finish in a major was tied second at last year's US PGA Championship, won by Justin Thomas.
Spieth, the victor in 2017, Kisner and Schauffele went into the final round at 9-under-par, with a two-shot lead, while Woods lurked four back. With another tough save from off the green at the 16th, he didn't miss a shot the rest of the way.
Hogan was tied for the lead after 54 holes in 1953 before going on to lift the Claret Jug in his only appearance in the event, but the last two winners, Lawrie and Padraig Harrington, trailed by 10 and six strokes respectively after three rounds.
Woods, wearing a smart blue sleeveless jersey that matched the colour of the summer sky, dissected the fairway with his opening tee shot, twirled his iron in his hand and strode off with a huge media posse in tow.
Tommy Fleetwood's five-wood approach to the 17th from a divot was pretty special, but Zander Lombard edges it after holing his approach to the 18th for an eagle two.
Tiger Woods of the U.S. walks on to the 18th green during the final round
He saved par with 8-foot putts on the 12th and 13th, two of the most pivotal putts all day.
Open runner-up Xander Schauffele insists he will look back on a chaotic final day at Carnoustie as a positive experience, despite missing the chance to win his first major.
Further trouble and another dropped shot was to follow at the next and it was with his head bowed deep in thought he strode down the par-three 13th where he managed to stem the haemorrhage with a par.
World number eight Rory McIlroy was an angry golfer after his British Open title charge stalled on Carnoustie's devilishly hard closing holes on Saturday.
"The beginning of the year, if they'd have said, 'You're playing The Open Championship, ' I would have said I'd be very lucky to do that".
Heading out in the penultimate pairing alongside Kevin Chappell, Kevin Kisner had his sights set on a maiden major.
Rose, who is ranked No. 3, came to the 18th hole needing a birdie to survive the cut. After leaving his uphill putt several feet short, Woods' just missed his par attempt inches right.
And the Olympic champion carried on where he left off on Saturday, taking advantage of benign conditions to fire seven more birdies to match the 64s recorded by Steve Stricker and Richard Green the last time Carnoustie staged the Open in 2007. "If I can just make the birdies and try to limit the damage tomorrow it will be interesting". "I think I'm capable of that".
McIlroy had struggled to make any headway but when he eagled the 14th he rocketed to the top of the leaderboard.
The oldest & most prestigious major, a trip to The Open is a must for every golf fan.