United were at Valley Parade on Tuesday evening for their rearranged fixture with the Bantams.
And another disciplined and competitive performance saw the Blues extend their unbeaten run to six games, with substitute Jordan Gibson almost nicking it with a stunnng shot deep into time added on.
United dominated the first half possession and they created the first chance of the day when Moxon put a cross in from the right flank. Armer got there at the back stick but he couldn't get the direction he needed and it dropped wide of the post.
The Bantams had a run of corners but Holy brought the spell of pressure to an end when he came off his line to claim it.
Midfielder Callum Guy started to see plenty of the ball and he picked out Edmondson on the edge of the area. A flick invited Edmondson to chase but the keeper was off his line to scoop it up.
Another searching United ball forward was trapped by Dennis and he shielded it before finding Senior in support. He shaped to shoot but his effort was charged down.
Good work from Banks at the other end created space, and he fed it for Chapman to let one go. It was powerful, but it was always high.
The visitors kept the home side in their own defensive third and Armer looped a deep corner for Dennis to challenge for. It had Lewis beaten but it dropped wide of the mark.
Armer was a constant outlet and he twice had deliveries intercepted as he probed on the overlap, but things tightened up with McCalmont and Mellish thwarted by blocks as both defences did their jobs.
United kept coming and Edmondson clipped a header towards Guy, who fancied the shot. It was sliced and Lewis watched it on its way.
Barclay then backed up the forward play before picking out Edmondson, but his header was straight at the stopper.
A wild rising effort from Chapman signalled the start of the second period but chances were few and far between as the Bantams probed without getting any purchase.
It became a real battle, but Andy Cook was close as he diverted a loose ball towards the nets. Feeney made a vital block that took the swirling attempt onto the bar.
The Cumbrians responded with a decent spell and a drive from Guy brought a free kick just 19-yards out when he was upended. Moxon found plenty of power but it was always too high.
Both sides wanted the win, and a snapshot from Garner wasn't too far away. Holy had to spread himself as Bradford broke, with Cook at tight angle as he tried to squeeze it under the dive.
Oliver was next to try his luck, but his header was misguided, and it was the Blues who nearly nicked it with the last kick of the game as Gibson ripped into space. He connected with an absolute thunderbolt that smacked against the base of the post before being smothered.
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Bookings
Bradford City - Oliver (79)
Carlisle United - Moxon (37), Barclay (56), Armer (90)
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Teams
Bradford City - Lewis, Halliday, Ridehalgh (Crichlow 69), Platt, Smallwood (c), Cook, Gilliead (Pereira 86), Banks, Chapman (Oliver 68), Stubbs, Clayton (East 68). Subs - Doyle (gk), Bola, Kelly.
Carlisle United - Holy, Barclay, Feeney (c), Mellish, Senior (Robinson 90+3), Moxon, Guy, McCalmont (Gibson 84), Armer, Edmondson (Garner 64), Dennis (Patrick 64). Subs - Kelly (gk), Whelan, Gordon.
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Match officials
Referee - Will Finnie
Assistant Referee 1 - Lisa Rashid
Assistant Referee 2 - Christopher Ward
Fourth Official - Richard Bartlett
Observer - Joe Copeland
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Attendance - 17,653 (1,208)