The Owls travelled to south London looking to stretch their undefeated run to a club-record 20 matches and facing a Palace side bottom of the table without a point.
But Murray, back in the Eagles team after injury, fired them ahead after just 30 seconds and secured a surprise victory seven minutes from time after Michail Antonio had equalised.
Murray's winner meant Dave Jones tasted defeat for the first time in 18 matches since he took charge of the Owls back in March.
It was lowly Palace who began like the team in form, going ahead before a Wednesday player had even touched the ball.
Wilfried Zaha's deep cross found Yannick Bolasie and when he drilled the ball back across goal from the left Murray was on hand to steer it past Chris Kirkland and into the net.
The Owls tried to hit straight back and Antonio forced Palace goalkeeper Julian Speroni to tip his curler over before Chris O'Grady failed to connect properly with a cross from Daniel Jones.
Lozano Rodri then missed a gilt-edged chance to level when he blazed wide from close range as the Palace defence waited for an offside flag that never came.
Speroni was called into action again before half-time, this time by one of his team-mate's as Mile Jedinak's defensive header was looping towards his own net until the Argentinian tipped it onto the top of the crossbar.
But Palace remained a threat and new recruits Bolasie and Andre Moritz exchanged passes before the Brazilian's shot was deflected into the arms of Kirkland.
However, five minutes after the break Wednesday drew level when Palace failed to deal with Miguel Llera's corner and Antonio bundled the ball into the net.
Zaha felt he should have had a penalty when his shirt was tugged by Lewis Buxton in the area but referee Rob Lewis was not interested.
Yet Lewis did point to the spot when Murray took a tumble under what appeared a fair challenge from Mark Beevers, only to change his decision after protests from Wednesday's players.
That change of heart should have inspired the Owls to push on for a winner, but instead Palace secured the three points in the 83rd minute.
Zaha twisted and turned down the right again and this time his cross was prodded in by Murray to condemn Wednesday and Jones to defeat for the first time in six months.