United and Bradford met in the FA Youth Cup for the second time in two seasons at Brunton Park on Thursday night.
The Bantams won a dramatic encounter with almost the last kick of extra time last season, but it was the young Blues who took the bragging rights just five days ago when a Dylan Garvey goal made the difference in a Youth Alliance Cup group game.
Bradford had successfully negotiated their way past Blackpool in round on this term, in what was another thrilling clash, and United had been made to work hard against Chester as they also made their way to this stage.
The quirk of this particular competition is that the draw for round three was made last Friday. A home tie with Blackburn Rovers lay in wait for the victors of this game.
United started on the front-foot and a shot from distance had to be gathered by Donaghy as it angled right at him.
It became end-to-end with half chances exchanged, and Donaghy was tested again when Bell let fly from distance. Bradford broke, and a tackle on Morris brought a free kick which Sukiennicki fizzed over the target.
The away side should have taken the lead when Roberts beat two to get into the area. He put it on a plate for Morris but the big striker lost his balance and his shot was poked wide.
Carlisle settled the nerves when a deep Watt corner was nodded back into the mix by Day. Fishburn had work to do but he used his strength to get his head on the ball and he wheeled away in celebration as it dropped over the line.
Dangerous set pieces from the accurate Palit had the visiting attack interested, but it was the hosts who nearly doubled the advantage when Roan Steele went on a run. His centre was met on the volley by Fishburn and Donaghy had to dive full length to tip it away.
Bradford pulled level on 39 minutes when Palit curled a corner to the back stick. Sukiennicki was completely unmarked and he left Robinson with no chance as he guided a deft header into the bottom corner.
That looked to be the score as the clock ticked into time added on for the half but a mix up at the back for the Bantams saw Adam Walton capitalise as he rifled a suddenly loose ball into the bottom corner.
The young Blues pressed high and sniffed out the second balls after the break and substitute Dylan Garvey earned his team a penalty when he cheekily checked back inside the box, only to be felled.
Referee Stephen Dial pointed at the spot and Lewis Bell showed a calm head as he drilled the spot kick into the top corner.
Bell was the width of the woodwork away from doubling his tally for the night when he rattled an excellent 20-yard free kick over the ball. It smashed against the bar with the stopper well beaten.
Another decent delivery, this time from Jack Ellis, as he sent one over on the run, found Fishburn arriving late. He connected firmly but his header was too close to Donaghy.
Bradford threw everything at it during the closing stages but they lacked composure in the final third and United were able to see the job through, with a third round home tie against Blackburn Rovers now to be played before 12 December.
Carlisle United - Fishburn (20), Walton (45), Bell (54 pen)
Bradford City - Sukiennicki (39)
Carlisle United - Robinson, Steele, Kilsby, Leslie (c), Barnett, Day, Ellis, Walton, Fishburn, Bell, Watt (Garvey 46). Subs - Simons (gk), Breeze (gk), Taylor, Hill.
Bradford City - Donaghy, Rowe, Norman, Palit, Bentley, Maumbe, Wilson (Youmbi), Sukiennicki, Morris (Pointon), Wood (c), Roberts. Subs - Ombang (gk), Ajibode, Ormerod, Thewlis, Tunkara.
Referee - Mr Stephen Dial
Attendance - BCD