The Owls got off to the best possible start by taking the lead after only five minutes as a cross from John Hills was converted at the far post by Chris Eagles, playing his final game for Wednesday before the end of his loan spell from Manchester United.
It was Wednesday's first goal in more than 630 minutes of football and was greeted with delight by the visiting supporters.
Second-half substitute Garreth O'Connor inspired a Burnley fight-back with Micah Hyde having an effort cleared off the line by Hills.
But it was Wednesday who extended their lead when central defender Graham Coughlan headed home a corner from Chris Brunt.
Burnley's consolation was a goal six minutes from time as Garreth O'Connor was brought down by goalkeeper Nicky Weaver and he squeezed home the resultant spot-kick for his seventh goal of the season.
But for the Clarets it was too little too late.