Swindon did manage to break the deadlock just before the half-hour mark though as Kasim looked for the target direct from another set piece, and this time managed to slot it into the top left corner.
The visitors needed a little over four minutes to double their advantage when Nathan Byrne blasted the ball in from outside the box.
Obika could have put the win beyond doubt for the visitors early in the second half when he attempted to head the ball in from a corner, but his effort was kept out by the post.
Swindon did find a third a few moments later though, as Byrne played the ball in for Andy Williams, who had just come off the bench, to all but seal the win.
The result lifts Swindon up to third in the table, while Chesterfield sit 11th.