Striker Sam Vokes put Sean Dyche's side ahead just before half-time with a calm finish.
But that lead lasted less than two minutes before Chris Maguire, who was denied a penalty late in the game, slammed home from close range to ensure the hosts dropped out of the Championship automatic promotion places.
The first real chance fell to the home side after 10 minutes, the bounce of the ball deceiving new Owls signing Oguchi Onyewu but neither Danny Ings or Keith Treacy could steer the ball in with the latter's effort deflected wide.
Ings' drive from distance six minutes later caused Chris Kirkland some concern but the Wednesday stopper did well to tip over.
Despite the Clarets' early pressure, Caolan Lavery came close to putting the visitors ahead on 19 minutes but his effort across goal evaded everyone and the far post.
Dean Marney had a good chance midway through the first half but he poked wide under pressure after a neat through ball from Vokes.
Connor Wickham called Burnley keeper Tom Heaton into action five minutes from the break, the former Cardiff City keeper having to be smart to tip a long-range effort round the post.
But, having soaked up some rare Wednesday pressure, it was the Clarets who took the lead three minutes before the break.
Vokes collected a Marney chipped pass over the top and slotted past the on-rushing Kirkland for his 13th of the campaign.
That lead lasted less than a minute as Maguire drew the Owls level, firing home from close range after Burnley failed to clear a Wickham free-kick.
After a frantic end to the opening 45, it was a more serene start to the second half, neither side creating a clear-cut chance until just before the hour mark when Lavery's deflected effort was somehow clawed away by Heaton.
That opened the game up and five minutes later Liam Palmer somehow got back to head Dave Jones' shot off the line after some great play by Vokes.
Scott Arfield had a chance to restore the home side's lead on 71 minutes but drove his shot across the face of goal having dispossessed Palmer.
Wednesday thought they had been awarded a penalty 11 minutes from time but referee Stuart Atwell booked Maguire for a dive, much to the anger of the travelling fans.
And Ings really should have secured all three points for the Clarets late on but he fired against the post and, as a result, Burnley slipped to third in the Championship after QPR's 2-1 victory over Huddersfield.