Robinson gave Derby the lead after nine minutes as he worked himself some space in the box before finding the corner of the net.
Derby failed to build on their early lead despite testing Lewis Price on a number of occasions throughout the first half as the keeper made impressive saves from Robinson and Jamie Ward.
Palace continued to struggle in the second half and Price performed more heroics as he denied Paul Green 12 minutes from time.
Callum Ball had the chance to put the game to bed with three minutes remaining but his strike from the edge of the area failed to hit the target.
Despite being wasteful in front of goal, Derby earned the 1-0 win to put themselves in the draw for the fourth round.