It was Cheltenham who took the lead after 22 minutes with a goal from Myrie-Williams from the edge of the box making it 1-0.
Cheltenham then doubled their lead on 24 minutes with a goal from Craig Reid.
Peter Brezovan had to make a couple more good saves to help keep the score down, and Swindon didn't look like scoring at all during the first half.
On 55 minutes Blair Sturrock pulled a goal back for Swindon from inside the box after a Cheltenham mistake.
Swindon pushed forward for an equaliser, but on 70 minutes fell further behind after the ball hit Ben Tozer in the box and went in for an own goal making it 3-1 to Cheltenham.
Swindon hit the bar late on with a header from Sturrock, but it was not enough and Swindon are now out of the Trophy for another season. Overall quite a poor game where Swindon although they had chances didn't look like winning. Swindon now have the weekend off and don't play again until Saturday 20th October when they play Tranmere at home.