Last week’s Scottish Open champion Chris Gotterup climbed into second 12-under through 15. Harris English and Wyndham Clark are a further shot back at 11-under.
Scheffler’s playing partner Li Haotong was at 11-under until a double bogey at the 14th saw him drop back in the pack.
Home favourite Rory McIlroy reached minus 10 for the tournament through the front nine, but his challenge stalled with a sloppy double bogey on the par-four 10th.
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Scheffler made three birdies in his first five holes to serenely reach 17-under par for the tournament, with his closest challengers at that point nine-under.
Scheffler holed long par putts on the sixth and seventh holes to stay in total control, but slipped up with a double bogey on the eighth after it took him four shots to reach the green.
It was the first time he had dropped shots since the 11th hole of his second-round 64 on Friday.
But when leading at the 54-hole stage, Scheffler has won his last nine tournaments and he showed why on the ninth with a trademark precise iron shot setting up another birdie.
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Li also birdied to keep his tilt at becoming the first Chinese man to win a major alive.
McIlroy hung on the fringes of contention with three birdies on the way out, against a lone bogey on the par-four fourth.
But he fired his approach shot from the rough on 10 through the back of the green and eventually made a six which effectively ended any hopes of a lifting second British Open title this weekend.
His playing partner Matthew Fitzpatrick, bidding to become the first English winner of the tournament since Nick Faldo in 1992, was 10-under after 11 holes, six shots behind.
Bryson DeChambeau soared up the leaderboard with a seven-under 64, the joint-best round of the championship, to finish on nine-under and will be left to rue a first-round 78.
“What I did for the past three days, I said, every time is go time, like you’ve got to come back from five, six back,” said the two-time US Open champion.
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“That’s the mentality I had.”
Shane Lowry, who won the title when the British Open returned to Portrush after a 68-year absence in 2019, ended with a 66 to finish on two-under.
