Bradford lead the shootout 3-2 after Peter Murphy, brought on with just a minute of extra-time to play for his spot-kick skills, fired over the bar. But Alan Rogers hit the post with Bradford's fourth penalty and Michael Bridges made no mistake to pull Carlisle level.
It was all up to Zigor Aranalde after Anthony Williams saved from Dean Windass and the Spaniard finished home in style.
Carlisle had lead the game from the 31st minute, when Kevin Gall's miscued shot turned out to be a fine pass for Derek Holmes to put them in front from ten yards with a stabbed shot.
But it took a fine reflex save by Williams to prevent Bradford taking a 28th minute lead, keeping out a firm 12-yard drive by Eddie Johnson, one handed.
Then Lee Holmes rattled Carlisle's post with a fizzing 25-yarder.
The floodlights on one side of Brunton Park failed six minutes into the second half and referee Mike Pike consulted both managers before taking the teams off for a nine-minute delay while the lighting was fixed.
After the resumption Bradford had the better of it as they battled for a leveller, Eddie Johnson shaving the post with a 25-yarder and David Wetherall heading wide from a corner, before Williams tipped an 80th minute Marc Bridge-Wilkinson drive over the bar.
But, a minute later, Bradford were level when Eddie Johnson met a dropping ball with a volley from the edge of the box that seemed to go through keeper Williams' hands.
And a dramatic, but goalless half-hour of extra-time, set up the penalty finale.