After weathering an initial burst of pressure, the Owls took the lead on 18 minutes following a break down the left by full-back Tommy Spurr who found Tudgay, his left-foot shot beat Burnley goalkeeper Brian Jensen comprehensively.
Wednesday should have increased their lead three minutes later when Spurr again set up Tudgay, but this time Jensen produced a fine reflex save.
Striker Clarke was to prove a handful throughout the entire 90 minutes for the Burnley defence and forced another two saves from Jensen before the half-time break.
Six minutes into the second half striker Steven Thompson missed a great chance to equalise, blazing a shot over the crossbar following a mistake by Mark Beevers.
But the home side did not have to wait long for their equaliser as Chris McCann headed a Robbie Blake free-kick beyond Lee Grant.
On the hour mark, Clarke thumped a volley against the crossbar with Jensen a spectator.
Wednesday went back in front when the Burnley defence failed to clear a free-kick from Richard Hinds and the ball broke for Tudgay to calmly beat Jensen.
Ten minutes later Brian Laws' team increased their lead when a mistake by central defender Michael Duff was capitalised upon by Clarke, who lashed a left-foot shot into the top corner of the net.
Three minutes later Wednesday scored again when Jermaine Johnson found Clarke, who ran half the length of the pitch before lifting the ball over the diving figure of Jensen.
With six minutes remaining Burnley reduced the deficit when substitute Chris Eagles rifled a free-kick into the top corner of the net.