The world number 10 started his second round one over par and six strokes behind overnight leader Kevin Kisner.
But the Briton carded the lowest round of the tournament so far to share the summit with Kisner and 2015 winner Zach Johnson on five under.
Fleetwood put the gloss on a fine 18 holes by sinking a nerveless birdie putt at the last, which has been playing as one of the hardest holes at the Angus links course.
Rory McIlroy enjoyed the briefest of spells in the company of the leaders, making a gain at the 14th before handing it back at the next hole.
Still, a second consecutive 69 send the four-time major champion into the weekend in good shape.
Tiger Woods endured an inconsistent opening eight holes, with two bogeys, two birdies and four pars, having hacked one stroke wildly into the crowd at the second.
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