Matt Heath’s own goal gave the visiting former football league outfit the lead inside 20 minutes.
Goalkeeper Phil Barnes was unable to stop Kristian Dennis’ second-half effort
Substitute Jake Speight pulled one back for the hosts late on in a lively cameo but Simon Weaver’s men failed to force a replay.
Without Dave McGurk (injury) and Simon Lenighan (suspension) from last Saturday’s 3-1 win at Leamington, the boss brought Leigh Franks and Luke Dean into his starting line-up.
Striker Callum Hassan retained his place up top alongside James Walshaw after scoring on his debut last week.
And the frontman’s muscular presence up front almost bore fruit within 90 seconds of the start when he nodded Leigh Franks’ cross wide.
Hassan was again in the thick of it to flick Craig Nelthope’s cross into a dangerous area and only the offside flag halted Adam Nowakowski heading Town in front.
Walshaw’s dangerous cross whisked across the face of goal but County’s Mark Lees was well-positioned to clear.
In their first attack, on 18 minutes, County broke the deadlock and it was Heath who was left exasperated in his own six yard box.
The ball broke free for County winger Richie Baker and his innocuous shot across goal was sliced into the top corner by the former Leeds United man from three yards.
The goal brought County to life in their forward movement. A superb save from Phil Barnes kept out Jamie Milligan’s curling free-kick from 25 yards before Kristian Dennis had a shot deflected narrowly wide.
A stale spell followed as County became used to defending Town’s direct style.
With half-time approaching Scott Duxbury caused a moment of panic in Town’s box with a mazy run but his deflected strike was acrobatically cleared by Nelthorpe.
Like the first half, the second began with Town’s height at set-pieces causing issues as Heath’s header and Nelthorpe’s deep cross were flapped behind by County stopper Danny Hurst.
However, with only two at the back and three Stockport forwards, Town were left exposed and visiting winger Chris Sharp almost finished off a brisk counter-attack only to blast into the side-netting.
Ideas were often found lacking in the home side’s forward play, and while Nowakowski fired a shot wide, goalscoring chances were of short supply.
County continued to leave three players up field when defending set pieces, and on one of their trademark counters, Dennis burst through the middle and struck a well-placed shot through Barnes’ dive. The Town stopper will have been disappointed not to have palmed the ball away.
With Town lacking composure and method, Weaver threw on striker Jake Speight to try and force a way back into it.
And the former Bradford City striker pounced on Hassan’s flick on to smack against the post before Walshaw tested Hurst’s reflexes from distance with a dipping shot from distance.
Young wingers Dominic Rowe and Jack Emmett were the manager’s final throw of the dice as Town reverted to 4-4-2 from their previous diamond structure.
The change brought a reaction and from a long bouncing ball, Speight found a yard inside the box to lift cleverly over Hurst and bring the CNG faithful to life.
He could have doubled his tally with two more chances soon after but his mistimed header was followed by a strong block on his right-footed shot.
But despite a late aerial bombardment, County held onto the win and progressed to the third qualifying round.