Tom Miller hit the extra-time winner to send Carlisle into the second round of the Carabao Cup after a 2-1 win at 10-man Fleetwood.
Miller had given the Cumbrians a 22nd-minute lead but Jordy Hiwula hit back in the 69th minute to send the tie into extra time.
Miller headed Carlisle ahead in the 101st minute before Fleetwood's Harvey Rodgers was then given a straight red card in stoppage time of the first half of extra-time and the hosts could not find an equaliser.