The visitors enjoyed more of the early possession with veteran midfielder Steve Watson testing goalkeeper Gabor Kiraly as early as the fourth minute. Just three minutes later striker Deon Burton shrugged off a challenge from James O'Connor but drove his shot wide.
At the other end an 11th minute corner by Robbie Blake was missed by Owls goalkeeper Lee Grant, but he was fortunate enough to see Clarets skipper Steven Caldwell head over the crossbar.
A long clearance by Kiraly midway through the first half was collected by Blake, whose pass found Jon Harley, but the makeshift midfielder dragged his shot wide.
Wednesday took the lead on 28 minutes with a penalty kick awarded by an assistant referee who adjudged that Caldwell had brought down Burton in the box.
But that was just the beginning of the drama as goalkeeper Kiraly took an age to get himself ready for the penalty kick, which enabled referee Scott Mathieson to eventually blow his whistle, a signal for Burton to stroke the ball into a virtually unguarded net.
But the home side's response was immediate as David Unsworth swung a cross to the far post where Wade Elliott headed the ball back across goal for Blake, whose header was kept out by goalkeeper Grant only for ex-Sheffield United striker Ade Akinbiyi to convert the rebound from close range.
The home side had the better of things in the second half with Blake creating an opportunity for Kyle Lafferty, only for the Northern Ireland international to curl his shot just wide from the edge of the box.
Caldwell then started and ended a move by getting on the end of a cross from Elliott but headed over from six yards.
Wednesday rarely threatened in the second half with Watson driving wide as the Burnley defence only half-cleared a corner, while the home side almost took all three points six minutes from time when Blake's corner was met by the head of Clarke Carlisle, only for Grant to ensure that Brian Laws' side collected a point.