The visitors had their first chance when Robbie Blake forced England goalkeeper David James into a save.
BBC: SPORT | Football | English Div 1 | West Ham 2-2 Burnley
But former Wrexham defender McGregor then sneaked up at the far post to head home Robbie Blake's cross.
Sensing Wigan were taking control, Bolton boss Owen Coyle brought on Robbie Blake and Rodrigo Moreno to replace Taylor and Petrov.
Steven Caldwell had earlier beaten Richard Dunne and Brad Friedel to a Robbie Blake free-kick to put the home team in front.
The former Manchester United winger then set up Robbie Blake for a header that was only inches off target in the 23rd minute.
That extended Ipswich's scoring record to 13 consecutive games but Robbie Blake equalised for Burnley from the penalty spot to sparking a late rally.
Another glorious chance came their way 10 minutes later when Thompson sprung the offside trap, raced down the right flank and squared the ball to Robbie Blake.
There were also newspaper reports that Laws had a bust up with striker Robbie Blake after he was substituted during the 6-1 defeat by Manchester City in April.
Burnley started strongly in the second half, Robbie Blake forcing a good low diving save from Cudicini, and later hitting a post with a rebound from an acute angle after the goalkeeper fumbled a free-kick.
The trend continued after the break - a deft Elliott backheel setting Robbie Blake free while Michael Duff came within inches of converting a Thompson header across goal - as the Blades struggled to cope with their opponent's superior invention.
It proved crucial for within 15 minutes of the second half Burnley made the game safe, with Little first combining with Robbie Blake in the 57th minute, scything his way through the Wednesday defence before curling a low shot beyond Pressman.
The Clarets, founder members of the Football League, played with freedom and ambition - and should have doubled their lead after the break but Nick Montgomery blocked Joey Gudjonsson's hooked shot while Kyle Walker made a superb challenge with Robbie Blake poised to strike.
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