Tranmere took the lead after 22 minutes when substitute Joe Thompson rose at the far post to head in an Abdulai Bell-Baggie cross.
The visitors had chances to double their lead in the first period with the best opportunity falling to Jake Cassidy, who beat the offside trap but Bartosz Bialkowski saved well.
A dominant Tranmere could have sealed it after 69 minutes when Zoumana Bakayogo broke away on the left and pulled the ball back for Adam McGurk, whose shot was saved superbly by Bialkowski.
County had the post to thank for keeping the scores level after 77 minutes when a firm shot from the angle by the lively Bell-Baggie struck the woodwork.
