A massive drive on the following hole nearly took him to the green and he duly went on to birdie that as well to pile the pressure on overnight leaders Kevin Kisner and Zach Johnson.
Sharma made three birdies and three bogeys on a day when conditions were most ideal to post a low score.
But Woods admitted he was happy with his round overall.
The last player to make a successful title defence was Ireland's Padraig Harrington, who coincidentally won at Carnoustie in 2007 and again 12 months later at Royal Birkdale.
Brooks Koepka is unlikely to be celebrating back-to-back major victories at the British Open on Sunday.
He's four shots back of Spieth and two others after a sparkling 5-under 66. He made six birdies and his only mistake was a bogey on the 16th, where he left his first putt from off the green short. He is even par overall.
The 24-year-old and a swathe of his fellow young Americans threatening to dominate the future of the sport will also have a returning Tiger Woods.
Woods shot a 66 on Saturday, and briefly saw his name sit atop the leaderboard following six birdies in his first 14 holes. "It's not that bunched", added the four-times major champion. A monster birdie on the ninth was followed by an approach shot on 10 that left his ball just a foot from the cup.
He got a lucky break at the last, where his pulled tee shot came within inches of landing in the Barry Burn.
Kevin Kisner now leads the Open Championship.
"He's really started to roll the ball well (on the greens), and he's hitting the ball exceptionally". All day long they had heard the cheers rolling across the links of Carnoustie and, finally, he was here.
Golf hasn't stood still in Tiger Woods' absence. It took him to five-under par for the tournament and inside the top-10 with leaders still out on the golf course.
But the three-time Open champion did endure some tricky final holes. "He's putting well too and heading in the right direction", said Norris.
It was the first time Woods had topped a major leaderboard since the Masters in 2013.
Carnoustie is still waiting on the one element that defines a strong test at the British Open: the wind.
Shortly after Woods finished his round, Italy's Francesco Molinari seized the clubhouse lead on six under par after a superb, bogey-free round of 65. He won past year, and he's playing great golf right now.
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.
But at least he's still playing. Others who won't be around for the weekend include Dustin Johnson, Justin Thomas, Sergio Garcia, Bubba Watson and Matthew Fitzpatrick. 1 and 2 in the world, missed the cut. He had successive birdies on the 14th and 15th before rolling in a long putt on No. 18 for a birdie that put him in the final group.
Johnson spoke to media shortly after home favourite Tommy Fleetwood and former world number one Rory McIlroy had held court in a packed interview room after they also pushed themselves firmly into contention for the title.
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