Coulson, a summer signing from Grimsby, arrowed in a close-range effort as goals from Daniel Parslow and Ashley Chambers sealed a deserved success despite Jon Nurse's effort for the Bees.
A seventh-minute counter-attack by the Minstermen was not properly cleared and Parslow fizzed a powerful effort into the bottom corner. From nothing, the visitors doubled their lead. Chambers was sent racing into the box following a lofted throughball and fired high into the net.
On the half-hour Barnet seized a foothold in the game. Andrew Yiadom embarked on a run down the right wing and his cross was expertly turned home by Nurse at the near post.
However, Coulson restored York's two-goal advantage on the stroke of half-time after controlling Chris Doig's free-kick on the edge of box and sending an unerring shot into the bottom right-hand corner as the visitors sealed a first win following their return to the Football League.
Nurse and Anthony Edgar both went close after the break but York withstood an early second-half blitz. Nurse was carried off on a stretcher with an apparent neck injury as the Minstermen held firm and picked up maximum points.
