The pacey winger was put through by a delightful through ball from substitute Matty McNeil and he fired home from 16 yards into the bottom left-hand corner.
The first half started slowly and it was not until the 14th minute that either team had a meaningful shot on goal.
Swindon winger Anthony McNamee then picked out striker Barry Corr with a cross from the left, but he put his shot wide of the right-hand post.
Stockport hit back through the superb Pilkington, who went past Swindon captain Michael Timlin before unleashing a shot from 20 yards that goalkeeper Petr Brezovan did well to tip over the bar for a corner.
The winger then set up team-mate Dave Mooney, whose first-time shot was well saved by Brezovan four minutes before the break.
Swindon's took the lead on the hour mark with a hotly disputed penalty. Defender Jim McNulty was adjudged to have pushed Simon Cox and the spot-kick was given despite protests from the whole Hatters team.
Right-back Jack Smith saw his penalty blocked by goalkeeper Owain Fon Williams, but the ball fell kindly back to him and he netted the rebound back into the bottom right-hand corner.
After Pilkington had equalised Stockport were thankful for goalkeeper Fon Williams when he pulled off a superb double save to deny Swindon all three points.
Cox picked out substitute Blair Sturrock 12 yards from goal and he looked certain to score.
Sturrock blasted the ball goalwards from 12 yards, but Fon Williams somehow managed to push the ball away before blocking the follow-up shot from Jon-Paul McGovern.