The two sides were locked at 0-0 in a stop-start encounter that saw both sides hit the woodwork during a feisty and entertaining first half.
Despite dominating the majority of the opening exchanges with Neil Harris forcing a good save from Owain Fon Williams in the first minute, it was County that had a good chance to open the scoring in the seventh minute.
Anthony Pilkington's drive deflected over Lions stopper David Forde and rebounded off the underside of the crossbar. Forde reacted well to smother Oldham's on-loan striker Craig Davies who latched onto the rebound.
Fon Williams produced a brilliant save low to his right to tip Marc Laird's precise header onto the post after a low driven cross from Chris Hackett.
Laird came close in first-half stoppage time when he blasted a vicious volley over the bar from 15 yards having brought Adam Bolder's hooked cross down well with his chest.
Harris came agonisingly close to getting the goal he needed to equal Teddy Sheringham's all-time record with two attempts in the space of three minutes.
The first, on 48 minutes, saw him lash a cracking left-foot effort over the bar having read Tresor Kandol's towering flick-on with Fon Williams well beaten.
Johnny Mullins then did well to hack Harris' toe-poke off the line with Fon Williams only able to parry Lewis Grabban's venomous low drive on 50 minutes.
Both sides threatened from set-pieces but it was 23-year-old Martin's venomous 25-yard strike that would prove decisive.
The Stockport wall parted like the Red Sea as his left-foot strike blasted through it and into the middle of the goal with unsighted Fon Williams rooted to the spot.
County were unlucky not to be awarded a penalty just minutes later when Grabban appeared to hack down Leon McSweeney who was set to pull the trigger inside the Lions box.
It was a scrappy finale with Kenny Jackett's men having to cling on as County piled balls into the box searching for an equaliser to save their unbeaten away record.