Swindon had more of the play in the first half, but failed to score with the score still being level at 0-0 going into the half time break.
In the second half it was Cheltenham who finally broke the deadlock and scored after 61 minutes with a goal from 8 yards out by Ben Gill.
The goal scored Gill was later sent off after picking up two yellow cards, but still Swindon couldn't make the most of only having ten men.
On 79 minutes Cheltenham doubled their lead with a close range header from Myrie-Williams which sent a lot of the travelling Swindon fans home early.
Overall a poor game where Swindon had chances mainly in the first half, but couldn't turn them into goals. Cheltenham only had a couple of chances and scored both of them.