Caretaker Manager Mark Cooper named a strong starting line-up with Tijane Reis, Ryan Harley and Ryan Mason all making a Town appearance on home soil for the first time.
After efforts from Andy Williams and Reis brought saves from Cirencester's Glyn Garner, Alex Pritchard opened the scoring just before the break.
Swindon started the second half with a different eleven and straight away Connor Waldon and Miles Storey were involved in the thick of the action, with the latter blazing over the bar.
Storey made amends later in the half when he lashed home from a half-cleared corner. Despite the home side pulling one back, Storey added a third for Swindon, slotting home after being sent clear.
