In the first half Swindon had most of the play, but just couldn't score in front of goal.
Sammy Igoe also come close to scoring with a long range effort, but it wasn't on target.
In the second half Swindon were still having a lot of the play, but Chesterfield were starting to get back into the game, with ex Swindon striker Wayne Allison coming close for Chesterfield.
Rhys Evans had to make a very good save to keep Swindon in the game, but Chesterfield were starting to have more chances, and it seemed only a matter of time before they would score.
On the hour mark Roberts was lucky not to get himself sent off with alate tackle on a Chesterfield player which started a on pitch fight. Both Roberts and a Chesterfield player were booked, when another referee might have decided to sent them off.
The only goal of the game came on 63 minutes when Wayne Allison scored with a close range header.
Swindon had some late chances to equalise after a number of corners, but they couldn't get any shots on target, with sub Rory Fallon hitting a good chance into the crowd.
Overall a fairly average game. Swindon really should have got at least a draw out of the game, but they never really looked good enough to do any more.
Swindon now start a run of four home games in a row starting with Tuesday's nights visit of local rivals Bristol City in the LDV Vans Trophy.