York led their opponents by just one point in the table before kick-off and took the lead thanks to Coulson's header after 41 minutes.
The Minstermen should have doubled their advantage after the break only for Ryan Bowman to miss an open goal but AFC Wimbledon failed to take advantage of the mistake.
Coulson threatened from an early York corner when he reacted fast to fire a shot at goal but Ross Worner was equal to it.
Danny Hylton had already seen one shot blocked and then fired wide after 31 minutes before Worner was required to tip another Coulson effort over the crossbar.
The Minstermen took the lead four minutes before the break when a corner was only half cleared and Lanre Oyebanjo lifted the ball back in for Coulson to head home.
The second half sparked into life when Coulson clashed with Sammy Moore and referee Brendan Malone was forced to step in to cool matters down.
York were denied a penalty when Josh Carson's shot appeared to be blocked by an arm before Bowman missed an open goal from six yards after 73 minutes following Worner's mis-kick.