A John Rooney double condemned Whites to a second consecutive league defeat.
Last season’s National League Player of the Year, who joined Stockport in the summer from Barrow, slammed home a second half penalty, before slotting in his second and County’s third late on.
Substitute Alex Reid had earlier doubled the home side’s lead as Athletic were unable to build on a first half clean sheet.
Andy Hessenthaler’s men, including centre-back and academy youngster Will Moses making his National League debut, defended resolutely throughout the opening 45 minutes and nearly stunned the home side when Steve Rigg slid and narrowly missed Ade Azeez’ vicious cross into the Hatters’ area.
It took the hosts until just before the break to have their first meaningful effort on goal as Connor Jennings mis-kicked twice within the space of a couple of minutes.
The home side made their breakthrough when another Dover debutant, goalkeeper Josh Bexon, upended Reid in the area, with Rooney making no mistake from 12 yards as he sent Athletic’s stopper the wrong way.
Reid volleyed over from the edge of the box, but the attacker was more precise with his next opportunity.
The striker converted Richie Bennett’s pass, and despite protestations from Dover for a foul on Will De Havilland in the build up, the goal stood, leaving Hessenthaler’s side with the challenge of scoring at least two goals.
That task proved insurmountable for Whites and though Rigg went close with a close-range header, it was Stockport who put the game beyond doubt at the end when Rooney fired Sam Minihan’s lay-off into the bottom corner.
Earlier, boss Hessenthaler made three changes to the side that started the 4-1 against Bromley on Tuesday night, with Bexon, Moses and Bilel Hinchiri replacing Yusuf Mersin, TJ Bramble and the injured Harry Ransom.
Dover are back in league action on Tuesday 13th October when they host Aldershot Town, kick off 7:45 pm.