Mark Cooper's side made all the early running with Jack Barthram side-footing wide of goal after being played in on the right and Massimo Luongo firing an effort off the foot of the upright.
Town were awarded a penalty when Jack Barthram was hauled down in the box but Michael Smith was denied the opening goal by goalkeeper, Shortt. Smith did make the breakthrough from the spot for Swindon shortly afterwards following a deliberate handball. This time, he made no mistake, blasting the ball high into the net.
The striker made it two before the break, steering home a low ball in from the left flank.
After half time, the home side reduced the arrears thanks to a stunning free-kick that arrowed in beyond the dive of Tyrell Belford.
A host of changes then followed for the visitors, a move which bore immediate fruit for Town. Substitute Will Randall turned in a cross from the right after some neat build-up play. The youngster would go on to grab his second of the evening when he was sent clear and rounded the goalkeeper before slotting into an empty net.
Having been denied by the Petersfield goalkeeper a number of times during the evening, George Barker got his name on the scoresheet before the end. Some clever skill on the edge of the box took him past two defenders before he lashed the ball into the net.
The home side did pull back a consolation late on through Ryan Coker.
