The American 2023 champion had appeared nearly untouchable as he started the day at seven under par, but struggled to find the fairway and showed little of his earlier finesse on the challenging Shinnecock Hills course.
Clark drove into the rough en route to a bogey on the par-three second and struggled with the sloped green for another bogey on the par-five fifth. He missed an easy putt for another bogey on the seventh, making the turn at four under.
Clark's implosion came just as Burns found his best level.
Having carded level-par through three rounds, he birdied the first, third and fifth holes before draining a near 50-foot putt for another on the par-four eighth. He missed a relatively simple putt on the next hole for bogey.
He was at three under par for the tournament with six holes left to play.
Clark's playing partner, world number one Scottie Scheffler, started the day at one under par in his quest for a career Grand Slam but was running out of time to make a move as two bogeys and a birdie left him at even par.