The 42-year-old began the day on level-par for the tournament but took full advantage of ideal scoring conditions to collect six birdies from his first 14 holes.
Woods, playing at his first British Open since 2015, rolled back the clocks on Saturday with a sizzling 66 - his best round at an Open since winning at a similarly parched Hoylake in 2006.
A victory would complete a most remarkable comeback for Woods, who has had four back surgeries, a marital scandal and a DUI arrest since his last major.
Birdies at the fourth and sixth took Woods to the turn in 34 and, with the leaders crumbling, gave the 42-year-old a one-shot lead over Molinari and Spieth, who double-bogeyed the sixth after hitting his second shot into a gorse bush.
"The way Francesco played today was attractive", said Woods after ending in a tie for sixth spot.
"I've had a chance to win".
Tiger's incredible run at The Open has effectively ended the biggest debate about his latest comeback - Can Tiger Woods win another major?
Spieth, who started the day in a three-way share of the lead with Kisner and Shauffele, crashed with a 76 to wind up equal ninth a further stroke back with USA countrymen Tony Finau (71) and last year's runner-up Matt Kuchar (72).
Bulls working to sign restricted free agent Jabari Parker to offer sheet
Parker is now a restricted free agent and the Bucks have extended him the one-year qualifying offer of $4.3 million. Parker, a Chicago native, has torn his ACL in his left knee twice during his career.
"I'm right there", he said. I figured today was, starting off the day, that nine was probably going to be the number.
"Given what happened last few years, I didn't know if that would ever happen again, but here I am with a chance coming Sunday in a major championship".
Woods, seeking his 15th major championship and first since 2008, leads by one stroke on 7 under from playing partner Francesco Molinari and Spieth. "I was very excited last night not to be down the road, ruing another Open opportunity gone. It's been a few years since I felt like this".
The way golf has been going, it would be reasonable to see the name "Johnson" atop the leaderboard and assume it belonged to the top-ranked player. I'd like to beat Gary Player one time. He set up birdie putts on three of first four holes but all missed, as did a par putt on the second and then there was a three putt bogey on the fifth.
Tiger Woods took the lead at one stage after a two-under front nine, but he faded over the last nine holes to finish at five-under-par, tied sixth. "I felt I really had control of the ball".
"I've been called Dustin many times", Zach Johnson said. "Even if it's windy, as long as it's not wet, I'll try and get myself back into it".
It was his best score in a weekend round of a major since the 2010 U.S. Open at Pebble Beach.