Lee Johnson’s side are looking to gatecrash the play-offs and they now stand just five points off the top six.
The match looked to be heading for a scoreless draw until the Tykes pounced opn 88 minutes,
And it was the in-form Sam Winnall - a reported £300,000 target of Leeds United - who broke the deadlock to take all three points back to Yorkshire.
Johnson said: “I think when you come to Swindon, you have to set out a plan. You’ve got to make sure they can’t break the lines and get in between you and they rarely did.
“At half time, we came in and we’d had 10 shots to their one. Really we should have taken one of them.
“In the second half we did get pegged back a bit for 15 or 20 minutes but still looked like we could break on them and score.
“I’m delighted for the lads because they’ve put in a good shift there. It’s their third game in a week. Another really tough game but our organisation was that good we earned ourselves an opportunity for Sam to score the winner.”