The Glovers twice fell behind but second half strikes from Lewis Wing and Alex Fisher was enough for the visitors to extend their unbeaten run on the road to four matches.
Losing captain on the night and the experienced head of Stuart Nelson at half-time due to injury, the Town boss was pleased with his side’s comeback.
Speaking to iFollow Glovers, he said: “They’ve got good players and they’ve got one or two that have got a wealth of experience.
“We knew going into the game that they had over 3,000 (EFL) appearances to our 1,000 and when you take Nelson out I think we were down to 600.
“From there, it’s down to the players to take responsibility, which they did, and again it was good for our supporters to travel home knowing our players gave everything.
“I call it an under-23s squad and I’m very proud of them.”
Wing’s third goal in Green & White and Fisher’s fourth in as many games gave the travelling faithful plenty to cheer.
Watching on from the sideline, Way was pleased with both goals at the County Ground.
“Both strikes are unbelievable finishes,” he said. “We threw Omar (Sowunmi) up there and took that gamble and it paid off. Swindon will be bitterly disappointed but we’re over the moon with the result.
“I’ve been saying to Wingy every day to just keep shooting. Been moments of frustrating for him where he’s had three or four shots in a game and they’ve gone 10 yards wide or 50 yards over the crossbar, but I said keep shooting and fair play to him, it paid off.”