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Palace had not scored since October 25 heading into the match, although Darren Ambrose ended that barren run midway through the second half with a wonder strike from 35 yards.
Federico Macheda levelled from the penalty spot for United to send the game into extra-time but Murray's 98th-minute header earned Palace a historic win and a place in the semi-finals.
Sir Alex Ferguson made 10 changes to the side that drew 1-1 with Newcastle in the Premier League on Saturday but, despite the overhaul, United still started with nine internationals in their line-up.
Dougie Freedman made five changes to the team that played out a goalless drew with south London rivals Millwall at the weekend, with the Palace manager having one eye on his side's Championship clash with Derby County on Friday night.
There was still room in the Palace side for 19-year-old Wilfried Zaha, however, and the Ivory Coast-born forward signalled his intent in the second minute, breaking with pace before slicing his shot high and wide from the edge of the box.
And Zaha had a penalty appeal waved away after 16 minutes after being bundled to the ground by Fabio, although replays showed the challenge took place just outside the box.
United struggled to find their feet but carved out their first opportunity after 20 minutes when Antonio Valencia exchanged passes with Macheda before finding Dimitar Berbatov, who put his shot wide.
Zaha continued to prove most likely to make the breakthrough for Palace but lacked composure in front of goal, shooting well over after receiving the ball on the edge of the box and turning.
United emerged from the break with renewed energy and Valencia sent a tame side-footed shot wide before Ravel Morrison, a half-time sub for Berbatov, forced a smart save from Palace goalkeeper Lewis Price.
Palace then took the lead against the run of play when Ambrose picked up the ball in the middle of the park and unleashed an unstoppable swerving shot past Ben Amos and into the top corner.
United struck back just three minutes later when Palace captain Paddy McCarthy hauled down Macheda, who dusted himself off to coolly slot home the spot-kick.
But United were unable to put their Championship opponents to the sword and in fact it was Palace who went closest to winning the game in the 90 minutes when Murray lashed a fierce shot into the side netting.
Darron Gibson saw his shot from distance drift inches wide of the post after the restart but it was Palace who made the breakthrough eight minutes into extra-time when Murray glanced home Ambrose's free-kick to earn Freedman's men a place in the last four.
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