The 26-year-old was nine under after 12 holes but a couple of missed short putts ruined his hopes of breaking 60.
Scott leads by a shot from compatriot Rod Pampling but Tiger Woods is 10 shots adrift after a level-par 72.
England's Paul Casey is on three under but Ian Poulter was disqualified for signing an incorrect scorecard.
Luke Donald is the only other Briton still in the event after fellow Englishman Greg Owen and Scot Richie Ramsay missed the cut.
Woods' round included three birdies but he also had a double-bogey six at the 17th and is now 10 shots off the pace.
"That's probably the best ball-striking round I can remember," said Scott.
"My ball-striking and distance control and everything was pretty much spot on, with the exception of the tee shot on 16 where he made his only bogey."
Woods, in contrast, was very disappointed with his form in the almost perfect conditions at Muirfield Village.
"I'm just not rolling the ball the way I want," he said. "These greens are so perfect. If you put a putt on line it should go in.
"I'm just not rolling the ball on line consistently. I have to make more putts, simple as that."
There was some good news for world number two Phil Mickelson, who withdrew during the first round with a sore left wrist.
Tests have revealed that there is no fracture, although the joint remains inflamed.
"My intention is still to play in Memphis next week but realistically the chances are 50-50. I don't want to jeopardise the opportunity to compete in the US Open," the American left-hander said in a statement.
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