It was the visitors who started encouragingly on a sun-drenched afternoon at Edgeley Park.
Billy Paynter rose well but headed wide after four minutes, then with 13 minutes gone the Hatters were saved by the woodwork.
Michael Timlin crossed from the left and Simon Cox's looping header bounced out and to safety off the crossbar.
The hosts hit back when they tested Swindon keeper Phil Smith twice inside a minute. First he beat Carl Baker's shot away, then got down to parry Jim McNulty's header.
Baker then fired wide from distance, before defender Michael Raynes volleyed over directly from Anthony Pilkington's corner-kick.
After dragging themselves right back into the game, Stockport went in front on 32 minutes with a well-worked goal. The ball was played into the danger area and Baker picked out Stephen Gleeson, who rammed home clinically from 15 yards.
Swindon got back on terms just three minutes after the restart. Jack Smith played a short pass to Easton and he hit home a great shot from just outside the Stockport box.
The Robins were now looking menacing and Timlin fired inches wide, then Cox tried his luck but also shot just past the post from 15 yards.
Late on Pilkington curled a free-kick just past the post for the hosts, while Lilian Nalis also came close to a winner for the visitors.