Fidel O'Rourke was on the scoresheet in the 80-minute encounter that saw Barry Lewtas hand valuable match action to 13 players.
Northern Ireland U16 international Conor Bradley made his first appearance for the club after receiving international clearance on Tuesday, starting the affair at right-back.
The hosts went close to opening the scoring with just seven minutes on the clock when Luis Longstaff produced a good turn inside the penalty area before unleashing a curling strike that was well stopped by the Wrexham goalkeeper.
Oscar Kelly ensured it stayed goalless with a superb save from close range to thwart the Welsh side after 15 minutes of play.