A late Matty McNeil strike was enough to kill off a resilient Tranmere side as County went into the second round of the League Cup for the first time in four seasons.
Jim Gannon named an unchanged side for the visit of Cheshire neighbours, Tranmere Rovers.
The County boss did make a change on the bench, though, with fit again Anthony Pilkington replacing Gregg Tansey.
The game was dominated mainly in midfield in the early exchanges. There were few chances of real note as both sides cancelled one another out.
The best chance of the opening exchanges fell to the powerful Calvin Zola whose header almost drifted into the back of the net, but for Logan scrambling the ball away.
County were nearly in front when Ian Goodison headed the ball just over his own bar, who was trying to man-mark the dangerous McNeil.
Logan almost fumbled the ball on the 19th minute when he caught a powerful shot from Tranmere's Greenacre, but he nearly dropped it again to allow in Zola. Luckily for County, however, the Irishman regained his balance and the danger was averted.
Logan pulled off a wonderful save to again deny Zola moments later, having been fed a deep ball by Chris Shuker.
Paul McLaren had an effort saved by the in form keeper moments later, as the game remained deadlocked.
County started to play some good football after this, with Griffin and Tierney dominating the left hand side and Taylor and Dicker playing the ball about in the centre. Elding and McNeil meanwhile looked hungry up front.
It wasn't pretty, but at half time it remained all to play for.
Second Half
Tranmere created the best chance of the opening exchanges when Steve Jennings low shot hit the inside post and bounced away for the County defence to avert the danger.
County created a good chance when David Poole burst into the box; Coyne saved his shot before Elding hit the rebound that Coyne touched away for a corner ball.
County thought they were ahead when McNeil got on the end of a Liam Dickinson cross but his low drive went agonisingly wide of the left hand post, to the disbelief of the County following.
Moments later Ben Chorley headed just over the cross bar from a few yards out.
Robbie Stockdale's speculative effort from 25 yards just glanced the cross bar as the game started to open up in the final 25 minutes.
David Poole had the chance to put County ahead on the 70th minute when he was played in by a clever McNeil touch, just one on one with the keeper he lost his footing and the chance was cleared.
Tranmere went close seconds later when Zola's diving header went just over the bar.
County went agonisingly close 13 minutes from the end when Tierney's free kick was headed on by McNeil and Elding's effort glanced just wide of the post.
Logan pulled off a wonderful low save to deny Steve Davies as the game started to enter its climax.
County's appeal for a penalty kick following a foul on Dickinson resulted in the decisive goal. The referee allowed County to play on and McNeil smashed the ball home from inside the area. The goal was paled slightly by an injury to man-of-the-match Poole who had been instrumental in setting up the effort.
