United were unchanged from their Tuesday evening home draw with Forest Green for their weekend trip to Valley Parade, with Mike Jones making his 500th career appearance as he took to the pitch.
The Blues were looking to make it three away wins in a row, with a mini-run of just one defeat in their last five outings giving them confidence for what is always a tough venue to visit.
Bradford created the first opening when Connolly feinted to cross before taking the ball away from Iredale. His delivery was cleared to the edge of the box and Cooke was disappointed to send a rushed volley well wide of the upright.
Another good run from Connolly saw him skip away from a challenge on the right wing before sliding a low pass into the box. Former-Bantam Knight-Percival was on hand to make an important intervention.
Carlisle showed their first hand of the day with an excellent break which came when Iredale raced in to steal possession on the halfway line. Olomola took over and his through ball invited Iredale to continue the run. A dinked cross into the box followed but it was nicked away from Thomas just as he looked to apply a finishing touch.
The away side were visibly lifted and they came again when McKirdy went on a sprint right through the heart of the defence. He dropped a shoulder and went for the shot but O'Donnell was able to gather it in.
More excellent work from Iredale saw him angle another teasing run across the face of the 18-yard area. It opened up in front of him, but he disguised the shot with a reverse pass through the back line. The predatory Olufela Olomola did the rest as he moved the ball before crunching it into the bottom corner to register his fourth goal of the season.
Bradford were rattled and both Thomas and Olomola asked questions of McKirdy when he opted to shoot, having done the hard work with a superb and incisive run. It had power but his 18-yarder was always heading wide.
Midfielder Mike Jones received a standing ovation from the Carlisle fans when he ran the length of the pitch to carry out important defensive duties as Connolly tried to get the ball back into the mix.
An unbelievable pass from Byron Webster set United on another attack when he spotted Sagaf alone and unmarked deep inside the opposition half. A heavy touch appeared to have taken the initiative away but Sagaf stuck out a boot to win it back. He then unleashed a powerful effort but O'Donnell was able to watch it fly harmlessly behind the target.
Good defensive work prevented a thundering goalbound volley from Vaughan from testing Collin, but the keeper was needed when Pritchard burst through as the clearance was sent back into the area. A firm one-handed save was helped on its way out of the area by Webster.
But the same man showed that he didn't need to be asked twice when he was once again played into space at a tight angle. He kept his fizzed shot low and Collin could do nothing about it.
Things got worse for the visitors just three minutes later when Bridge stayed shoulder-to-shoulder with Pritchard as they tussled for the ball. The Bradford man went down and the referee was very quick to show a second yellow and give the midfielder his marching orders.
The one-man advantage was made to tell on 40 minutes when a Palmer free kick was whipped towards the back post. Mellor showed strength to get enough on his header to guide it over the line.
Nathan Thomas almost took O'Donnell by surprise with a curling shot which dropped towards the top corner, and Iredale put another good cross into the six-yard box as the ten men did everything they could to get back into the contest.
There was real controversy on 50 minutes when prolonged Bradford celebrations suggested that they'd scored their third goal of the day. The cross from Wood brought a close-range header from Clayton Donaldson, but Adam Collin pulled off a wonder save to keep it out. The ball eventually squirmed over the line but United's keeper was up immediately to argue that it had been scooped into the net by a hand. The referee signalled a goal, the home contingent celebrated, but the asisstant referee asked for a discussion and a free kick was given.
Carlisle dealt well with a number of deliveries from left and right as Bradford sent men forward, so Cooke took things into his own hands when he went for it from distance. It had plenty on it but it dropped over the bar.
More controversy came when former Carlisle youth player Ben Richards-Everton was given a straight red card as he threw Iredale to the floor, just moments after Vaughan had seen a volley saved by Collin and a follow-up from Donaldson had smashed back off the angle of post and bar.
The Cumbrians were a whisker away from an equaliser on 65 minutes when Thomas bent a great free kick into the mix. Webster got a slight touch, and that brought a full length save from O'Donnell. Knight-Percival was onto the rebound but another block sent his prod behind.
A swiped volley from Cooke went into the stands as chances started to dry up, but Bradford were gifted an opening when a loose pass from Elliott was fed to Connolly. He scuffed his shot and that made it easy for Collin to deal with.
United were a matter of inches away from an equaliser when a patient build up allowed Thomas to ghost beyond the back line. He took his touch well and O'Donnell was thanked as he got fingertips to a low shot which bent across the face of goal. The Carlisle winger was left with his head in his hands as it bobbled off the surface and shaved the outside of the post.
Sorensen, Thomas, Olomola and Hope all chipped away through the closing stages but it was the Bantams who sealed the deal when a break from a mishit Sorensen volley left Ismail with just Collin to beat. He rifled it in and it was the hosts who claimed the three points.
Bradford City - Pritchard (30), Mellor (40), Ismail (90)
Carlisle United - Olomola (14)
Bradford City - Richards-Everton (62 - straight red), Connolly (78), Cooke (90)
Carlisle United - Bridge (11 & 33 - sent off), Knight-Percival (39), Iredale (54)
Bradford City - O'Donnell (gk), Mellor, P O'Connor, Richards-Everton, Pritchard (Oteh 83), Donaldson, Vaughan (c) (A O'Connor 70), Palmer, Connolly (Ismail 78), Wood, Cooke. Subs - Hornby (gk), McCartan, Henley, Devine.
Carlisle United - Collin (c) (gk), Sagaf (Sorensen 82), G Jones, Webster, Knight-Percival, Iredale (Hope 73), N Thomas, Bridge, M Jones, McKirdy (Elliott 37), Olomola. Subs - Gray (gk), Mellish, Branthwaite, Loft.
Referee - John Busby
Assistant 1 - Ian Dudley
Assistant 2 - Stephen Wade
Fourth Official - Joe Simpson
Attendance - 14,217 (828 away fans)