Brian Gay, who turned in a 69, is alone in third place another stroke back while J.J. Spaun had eight birdies in a round of 62 to leave him in fourth place at 13-under 199.
Overnight leader Cook got off to a great start with birdies at the first two holes before making a clutch par at the third.
"We got a few putts to go early and just kind of kept the course," he told PGA radio.
"It could have gotten ugly pretty quick if I didn't get the ball quite up that hill," Cook said after a poor tee shot on three. "But we were able to hit it to about eight feet and make the putt and keep the momentum."
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Two more birdies, at the seventh and ninth, took him to the turn at 4-under but he stumbled at 14 with a three-putt bogey. Cook did not allow his first dropped stroke of the week to get to him, however, as he hit back right away with another birdie.
"I've grown a lot maturity wise, mentally," Cook said. "Nowadays I'm trying to work really hard to let things slide off my shoulder."
Kirk, who won the tournament in 2013, kept the pressure on with seven birdies and a bogey.
"It's nice to be back in contention," he said.
(Reporting by Gene Cherry in Salvo, North Carolina; Editing by Peter Rutherford)
