Massimo Luongo added a third for the visitors while the hosts had Liam Noble red-carded early in the second half.
Manager Mark Cooper was forced into one change to the starting line-up. The injured Harry Toffolo made way for Ben Gladwin to start on the left while there were also places on the bench for Matt Jones and Curtis Da Costa. Jon Obika and Andy Williams continued their partnership up front.
It took the latter just six minutes to open the scoring at Meadow Lane. Played in on the left-hand edge of the box, Williams struck a left-footed effort that rolled past Roy Carroll and into the far corner of the net.
Carroll would be on hand to deny the Town a second goal midway through the half: Massimo Luongo scampered clear of the last man and poked an effort goalwards but the big ‘keeper managed to get his leg across in time to divert the shot wide for a corner.
Minutes before that, the Australian midfielder had weaved his way into the box before firing across goal and wide.
With Foderingham largely reduced to making routine saves to keep the home side at bay, Obika almost doubled Swindon’s lead at the other end. Bursting through the middle, he manoeuvred the ball around the advancing Carroll and sent a shot goalwards only to look up and see a defender scurry back onto the line to clear the danger and stop it from hitting the back of the net.
Before the first forty-five was over, Swindon created another opportunity on the counter-attack. This time spearheaded by Nathan Byrne, the wing-back squared the ball for Ben Gladwin. The midfielder strode up to the rolling ball and arrowed an effort just over the angle of bar and post.
The fifteen minutes the other side of the break saw Swindon take control of proceedings. After withstanding a period of Notts County pressure which included Jack Stephens clearing an effort away to safety off the goal-line, the visitors doubled their lead. Ben Gladwin tied up his full-back in knots with some clever trickery before firing an effort across goal and picking out Andy Williams, who applied the finish at the far post.
Moments later, Liam Noble was shown a straight red card after an aerial challenge with Nathan Thompson.
Swindon pressed home their man advantage less than a minute later by notching a third goal. Gladwin was the creator once again, drifting away from his defender before picking out the advancing Luongo. The Australian duly obliged by smashing home from six yards.
Town continued to dominate proceedings. Before the game was over, Andy Williams could have notched a hat-trick but was denied by Roy Carroll before the veteran 'keeper did magnificently to keep out the follow-up from Massimo Luongo.