The visitors looked on course for a rare away win - their first since December 1 - after two goals from Steve Morison cancelled out Murray's first-half opener.
But the country's top scorer had other ideas and his second of the afternoon made it 18 home matches without defeat for the promotion-chasing Eagles.
Palace went ahead in the 27th minute when Jonathan Williams floated a cross to the far post.
Murray sneaked unnoticed between two Leeds defenders and planted a free header into the roof of the net from a couple of yards out.
Stephen Dobbie then fired over from a good position, before Leeds grabbed their equaliser.
Luke Varney broke down the left and his cross was missed by the Palace defence and found Morison for a close-range finish.
Selhurst Park was stunned in the 69th minute when the Eagles allowed a Kenny punt up field to bounce into the path of Morison. The Wales international tucked the ball past Speroni for his second of the afternoon and his third since joining Leeds from Norwich.
But Palace were not going to give up their unbeaten record without a fight and, with seven minutes, remaining Murray arrived at the far post to lash in Damien Delaney's cross and celebrate another goal.
- Sports & Recreation
- Steve Morison