Derby set up a Carabao Cup second-round meeting with Barnsley after beating 10-man Grimsby Town 1-0 at Blundell Park.
Matej Vydra opened the scoring from the penalty spot after 53 minutes and his goal proved enough as the first-round tie was completed at the second attempt.
Former Rams defender Ben Davies was later given his marching orders against his old club as the home side received their fourth red card in as many games this season.
James McArthur’s second half double was enough to seal Crystal Palace’s progression to the next round as they beat Championship high fliers Ipswich Town.