The Red Dragons had twice hit the bar in a storming second half before Simon Spender turned the ball into his own net as he tried to keep out James Walker's right wing shot following good work by Jay Smith.
Neither side could gain an advantage in a fairly lifeless first half when a nervous home side failed to make any great impression on County's tough defence.
Wrexham were out of luck when Ryan Valentine's cross was headed on by Lee McEvilly and Chris Llewellyn's header hit the bar.
Another Wrexham chance went begging when Spender picked up a loose ball on the edge of the box and Matt Somner blocked the shot.
Spender wasted another opening aiming a speculative long shot straight at keeper Kevin Pilkington.
Wrexham stepped up the pace after the interval when Matt Done skipped past Gary Silk and Smith only to give the goal keeper an easy save from a tame shot.
McEvilly was also out of luck when he curled a left wing shot beyond the far post. Llewellyn missed another opening, directing a low shot off target.
County went close to taking a lead when Myles Weston crossed and Valentine headed out over his own bar.
County went all out for goals, using four strikers, bringing on top scorer Jason Lee to help their cause.
A free kick from McEvilly went over the County cross bar and Jeff Whitley rattled the post with a drive from 20 yards but Wrexham could not get the goal they so badly needed.