Swindon fought back to earn a share of the points at Edgeley Park.
The visitors started the brighter of the two sides and Simon Cox saw an early looping header bounce away to safety off the Stockport bar.
The hosts battled back and went close through Carl Baker and Jim McNulty before Stephen Gleeson fired them ahead from inside the penalty area.
Swindon levelled after the break when Jack Smith found Craig Easton and he smashed the ball home from 20-yards.
Stockport: fon Williams, Mullins, Owen, Raynes, McNulty, McSweeney, Gleeson, Blizzard (Thompson 70), Rowe (Turnbull 82), Baker (Dicker 46), Anthony Pilkington.
Subs Not Used: Rose, Vincent.
Booked: Gleeson.
Goals: Gleeson 32.
Swindon: Phil Smith, Amankwaah, Morrison, Aljofree, Jack Smith, McGovern, Easton, Nalis, Timlin (McNamee 90), Paynter (Corr 63), Cox (Peacock 86).
Subs Not Used: Brezovan, Casal.
Booked: Aljofree, Amankwaah, Jack Smith.
Goals: Easton 49.
Att: 5,536.
Ref: Keith Hill (Hertfordshire).
Swindon Town manager Maurice Malpas told BBC Radio Swindon:
"It's amazing that away from home we seem to look solid and at home we're the complete opposite - this was a typical away performance by us."
"We were dogged - at times it wasn't pretty, at times it was pretty - but we looked far stronger as a team defensively, a point's a good result and 1-1 was probably the correct result."
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