The result left Palace 12th in the standings on 16 points from 14 games while Leeds moved two places up to 15th on 15 points after they ground out their third league win of the season.
Raphinha stroked his penalty past visiting goalkeeper Vicente Guaita, with the spot-kick given following a VAR check after centre back Marc Guehi handled a corner.
After a dull first half devoid of any chances, Leeds' Rodrigo dragged a Daniel James cross from the left well wide of the near post in the 53rd, as he failed to turn and get a shot on target.
Christian Benteke then missed a pair of sitters for Palace, heading wide at the far post before home keeper Illan Meslier scrambled to smother the striker's feeble close-range shot on the line.
Those misses proved costly for the visitors after the ball hit Guehi's outstretched arm deep into stoppage time, with Raphinha making no mistake with the penalty to delight the home fans at Elland Road.