County went ahead in the 20th minute with a sublime move started and finished by Adam Griffin.
His header from the edge of the six-yard box was the culmination of some terrific passing from Paul Turnbull and Wes Fletcher.
Even before the Hatters took the lead, they had created the better opportunities. Griffin almost put the home side in front after six minutes and David Poole fired a glorious chance over the bar.
Against the run of play, Posh equalised just eight minutes after Griffin's strike when an innocuous ball from Lee Tomlin, which should have been dealt with by Matt Glennon in the Stockport goal, was headed in by Aaron Mclean.
The expected collapse from County did not materialise as their defence, which had conceded 10 goals in their previous two league matches, held firm against an unusually subdued Posh attack.