Jon-Paul McGovern scored the winner for Danny Wilson's men in the only game in League One to survive the big freeze and the victory pulled Town away from the relegation zone.
Swindon took the lead in the 24th minute in stunning fashion. Good work from McGovern down the right-hand side found Charlie Austin unmarked in the box. The red-hot Robins striker let fly with a sweetly struck volley that found the top right-hand corner of the net - his 13th goal of the season.
Wednesday equalised five minutes later in controversial circumstances. Lewis Buxton ran into the penalty area and looked to have rounded Robins goalkeeper Phil Smith from six yards out to score.
But Smith brought down Buxton for a certain penalty and red card. However, the referee waved play on and Clinton Morrison was there to tap into an empty net from a yard out.
Swindon's winner came from a long throw by Michael Rose which was flicked on by the towering Sean Morrison in the box and McGovern was on hand to guide it home past Nicky Weaver's outstretched hand.