Cureton pounced eight minutes after the break following a mistake from Chesterfield keeper Rob Beckwith and he added his second, and seventh of the season for the Robins, with a coolly taken finish from Gareth Whalley's through ball.
Despite dominating throughout Swindon could have gone behind after five minutes when Chesterfield's full-back Alan O'Hare saw his header clear off the line by Jack Smith.
The home side slowly but surely began to exert their influence on the game and should have gone ahead on ten minutes. A quickly taken corner by Cureton found strike partner Trevor Benjamin, but somehow he directed his header high and wide from six yards.
Two minutes before the interval Benjamin was guilty of another missed opportunity after he was put through by Cureton only to lift his attempted lob wide of the Chesterfield goal.
A change at half-time saw on-loan Watford midfielder Toumani Diagouraga introduced and the Robins matched this with an increased attacking spirit.
A misplaced back pass by Aaron Downes went straight to Cureton but Beckwith was able to make the save, while two minutes later and Jack Smith was guilty of the worst miss of the game as Benjamin teed him up only for the full-back to slice his effort wide.
Minutes later, though, Swindon were ahead. Benjamin cut in from the right-hand side of the area and hit a left-foot shot which appeared to be going wide, but Beckwith attempted a save and as he did so fumbled the ball straight into Cureton's path for the diminutive striker to finish.
Cureton doubled his and Swindon's tally minutes before the final whistle after the impressive Whalley had put him through.