Robbie Blake took advantage of a poor clearance to fire Burnley into an early lead with a low shot on 14 minutes.
Blake and Kyle Lafferty tested Owls keeper Lee Grant, but at the other end the hosts spurned good opportunities.
Minutes after the break Blake stole the ball off Frank Simek to tee up Chris McCann, and despite a late home rally there was to be no Wednesday comeback.
Burnley manager Steve Cotterill:
"The back four and goalkeeper Gabor Kiraly were the foundation of the win. "I thought the back four was the best they've been as a unit this season and to be fair, at any stage, they've had dangerous forwards on the pitch.
"And that's the best Robbie has played. It's no coincidence he's scored, that's the hardest he's worked and he deserved his goal," said boss Cotterill.
Sheff Wed: Grant, Wood, Watson (Hinds 46), Simek, Spurr, Lunt, Whelan, Esajas, Small, Sodje (Burton 60), Jeffers (Tudgay 60).
Subs Not Used: Burch, Johnson.
Booked: Simek, Whelan.
Burnley: Kiraly, Duff, Carlisle, Caldwell, Jordan, Alexander, Elliott, McCann (Gudjonsson 73), Lafferty (Spicer 82), Gray, Blake (Akinbiyi 83).
Subs Not Used: Jensen, Unsworth.
Booked: Jordan, Alexander, Lafferty, Carlisle.
Goals: Blake 15, McCann 48.
Att: 18,359.
Ref: Paul Taylor (Hertfordshire).
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