The injury-ravaged Hillsborough team, without half their side, had struck from the penalty spot in the 67th minute after Chris Brunt and Burton O'Brien had both hit the woodwork.
Jon Harley was the villain, upending full-back Frankie Simek to allow striker Steve MacLean to grab Wednesday's first goal of the campaign, before Clarets substitute James O'Connor hit back in the 82nd minute.
Wednesday started in lively fashion. Brunt, again playing in an accustomed role up front, found space on the left and turned cleverly to drill in a curling cross, which was safely cut out by Brian Jensen as MacLean waited to pounce.
A Graham Coughlan free-kick minutes later almost found a way through - MacLean lunged but just failed to reach the ball from six yards as Jensen tidied up.
Former Sheffield United striker Andy Gray was foiled after he connected with an Alan Mahon corner, seeing his close-range shot hacked away.
And minutes later the lanky frontman stretched to meet a pinpoint cross from Mahon but steered his header over from eight yards.
Wednesday caught the Lancashire side on the break, with O'Brien inches from breaking the deadlock.
The former Livingston player surged from midfield to unleash an eye-catching strike, his left-foot shot from 25 yards beating Jensen but smacking against the bar.
Wednesday's Aussie keeper Brad Jones, on loan from Middlesbrough, foiled Wade Elliott by moving quickly to cut out his shot on the turn, before the hosts hit the woodwork again.
Brunt ran on to an O'Brien pass on Wednesday's left flank and produced an angled drive from long range, the ball flashing past the helpless keeper before crashing off the bar.
O'Brien was handed a half-chance in first-half injury-time as he ran onto a Glenn Whelan cross, but from eight yards out he glanced his header wide of the target.
Jensen produced a stunning stop on 46 minutes after Burnley failed to cut out Kenny Lunt's precise corner. Algerian centre-back Madjid Bougherra charged in to connect with his right foot, but saw his volley pushed away by the sprawling keeper.
MacLean finally had the ball in the net minutes later only for referee Colin Webster to rule it out for pushing in the area.
Hard-working Wednesday finally snatched the lead after Harley hauled down Simek in the box. MacLean stepped up and coolly slotted the spot-kick into the right hand corner, sending Jensen the wrong way.
O'Connor grabbed the equaliser in the 82nd minute, slotting the ball through Jones' legs from six yards after Frank Sinclair's cross was headed back by Chris McCann.
Burnley ended the game with ten men after Gray picked up a second caution in injury-time.