The Republic of Ireland midfielder scored the only goal of the game to ensure Steve McClaren's side made a winning start to their Championship campaign against Rotherham United on Saturday and doubled his tally two days later.
Hendrick headed home from close range in the second half and Chris Martin added a second in time added on as League Two outfit Carlisle were consigned to another defeat at Brunton Park after losing to Luton Town on Saturday.
Derby, beaten in the Championship play-off final by QPR in May, really ought to have won by a more emphatic margin but a gritty showing ensured they progressed to the second round.
Carlisle goalkeeper Mark Gillespie had to be alert to palm away Ward's downward header and the former Sheffield United man fired the rebound just wide of the far post.
Martin's left-foot strike hit the crossbar two minutes before half-time as Derby continued to probe without reward, then the Championship side were indebted to Lee Grant 59 minutes in when he produced a fine save to tip David Symington's venomous free-kick onto the crossbar.
The impressive Gillespie made another fine save down to his right to deny Leon Best a first Derby goal, but Martin sealed the victory two minutes into stoppage time with a clinical right-footed finish.
