Adam Griffin's shot was touched by Anthony Elding to send it past keeper Anthony Williams in the 58th minute.
The home side had a great chance to equalise in the 75th minute when Michael Raynes handled, but Ryan Valentine's spot-kick was easily saved by Conrad Logan.
Logan was County's hero as he thwarted a bright second half start from Wrexham by somehow keeping out a point-blank header from Neil Roberts.
Wrexham had little chances after that and it needed a goal-line clearance by young defender Wes Baynes to deny Anthony Pilkington a second County goal near the end.
Wrexham boss Brian Little must really bring in some fresh faces if he has any chance of pulling the Red Dragons out of problems.
They had little to offer in attack and their low confidence was all too evident, particularly against a rock-like County rearguard, who wasted several break-away chances in the first half.