The 42-year-old began the day on level-par for the tournament but took full advantage of flawless scoring conditions to collect six birdies from his first 14 holes.
Kevin Kisner, who had begun the day on nine-under with Spieth and Schauffele, suffered a terrible start with a double bogey at the second and a bogey at the third to slip back.
Only five strokes separated the top 19 players on the leaderboard going into Sunday's conclusion with four-times major victor Rory McIlroy in the mix as are Justin Rose and Tommy Fleetwood who hope to become the first English player to win the Open since Nick Faldo in 1992.
On a day when Woods electrified the gallery with a 66 that pulled the 14-major victor within four strokes of the lead, Spieth, with three majors already, showed again that he is the man to beat of the next generation.
The 35-year-old was paired with fellow Nike athlete Woods, and broke out of a five-way tie for the lead on the 14th hole, tapping in for a birdie. "It's been a few years since I've felt like this", an ebullient Woods said after his best score in a Major since another 66 in the second round of the 2011 Masters.
Asked about becoming the first Englishman to win a British Open in 26 years, Fleetwood said, "It would be very special".
His previous best finish in a major was tied second at last year's US PGA Championship, won by Justin Thomas.
Spieth began with an eagle two at the first and added four more birdies to get to 9-under-par.
"Walking up, I thought I was in big trouble, no swing, right in the trees, based on the way the crowd was moving around the tree", Spieth said. Lyle's unbroken run started at Turnberry in 1977, when he also played as an amateur, before returning at St Andrews the next year in his first as a professional.
"I didn't want to be too far back if the guys got to 10-under par today".
He ended the day on five-under 208 while compatriots and overnight leaders Zach Johnson and Kevin Kisner had not completed their rounds by press time. Instead of being two shots behind, he was four back, certainly not out of it. "I really felt like I had control of the golf ball today".
Playing alongside the resurgent Woods in the final round, Molinari carded a nerveless two-under-par 69 to win the Claret Jug by two shots from McIlroy (70), English hope Justin Rose (69) and Americans Xander Schauffele (74) and Kevin Kisner (74).
"I'm playing golf for me now".
That two-putt from 30 feet on the par-five 14th meant Woods was one of seven co-leaders at that point, although a bogey on the 16th forced the American to settle for a five-under 66.
"As long as 18 is downwind, I don't really care", he said.
"I picked up where I left off last night", Rose said. "Maybe my wedge play wasn't quite as good as it should have been, but I gave myself plenty of chances".
Rose's round matched the lowest score shot in an Open Championship at Carnoustie, a traditionally hard course on Scotland's east coast.
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