Blues striker Geoff Horsfield had cancelled out Iwan Roberts's opener on a tense afternoon that ended in tears for Norwich - who will have to wait a little longer to make it back into the top flight.
Birmingham had began the game under the roof in Cardiff well and they were nearly in front inside the first 15 minutes but Stern John, latching onto Bryan Hughes's pass, dragged his effort wide when through on goal.
Despite a slow start, Norwich soon found their feet however, and David Neilsen almost prodded home the opener on 18 minutes following a mix-up in the Blues defence between Michael Johnson and keeper Nico Vaesen.
Back at the other end, the lively Horsfield felt he should have had a penalty shortly before the half-hour mark when he fell under Malkay Mackay's challenge, but referee Graham Barber waved away the protests.
There was then drama at both ends late on in the first half. First Clint Easton fired wide after an excellent run by Gary Holt had opened up the Birmingham defence, while Horsfield was denied by a superb reaction save from Norwich keeper Robert Green as Birmingham looked for the advantage at the break.
Both sides kept up the quick pace for much of the second period with Paul McVeigh going closest for Nigel Worthington's side when he brought a finger-tipped save from Vaesen.
As the tension grew and extra-time loomed, Paul Devlin was only inches away from snatching victory for Birmingham when his free-kick spun narrowly over after Easton was judged to have handled just outside the area.
Olivier Tebily then lashed the ball over from a corner as Blues pressed but there was to be no breakthrough in normal time.
That did come, however, just two minutes into the extended period when substitute Roberts headed Alex Notman's cross from the right back across goal to leave Vaesen standing, and send Norwich into temporary ecstasy.
The delirium lasted just nine minutes, however, as Horsfield powered home the equaliser from close-range after John had headed down Kenna's right-wing cross.
Michael Johnson then saw his header come back off the post for Birmingham with just three minutes left on the clock but the final was to be decided on penalties.
And, after misses from Mulryne and Sutch, Carter stepped up to blast home to consign The Canaries to another season in the First Division.