The 18-year-old Brummie kept a cool head to slot home a fourth and clinching penalty to win the shoot-out in a thrilling play-off final.
Blues held the initial superiority with Martin Grainger driving a 20-yard free-kick over the bar and Bryan Hughes shooting wide when well-placed.
Such was the intensity of Blues' early pressure it was 13 minutes before Norwich mustered their first attack, with Mark Rivers' cross deflecting off Grainger for a corner.
Stern John broke clear on 14 minutes to screw his shot wide with only goalkeeper Robert Green to beat.
As Norwich gained the initiative, hesitation between Michael Johnson and goalkeeper Nico Vaesen allowed David Nielsen to nip in and toe-poke an effort just wide.
Gary Holt then popped the ball up for Rivers to cross for Clint Easton who volleyed wide.
In first-half injury-time Norwich keeper Green produced a stunning point-blank save from Horsfield's powerful volley.
Norwich responded and Vaesen was forced to tip over from Paul McVeigh who then turned provider, crossing for Nielsen to head wide from 10 yards.
Adam Drury slid in to block John's shot and when Easton was harshly called for handball by referee Graham Barber Paul Devlin fizzed a shot inches over the bar.
Blues mounted a strong finish and Tebily spurned a half-chance when he fired over from six yards out.
In injury-time, Blues skipper Jeff Kenna saved an almost certain goal as he intercepted substitute Alex Notman's cross just ahead of Iwan Roberts.
Roberts was on target two minutes into extra-time, perfectly directing a powerful header home after Grainger's slip gave Notman the chance to cross.
Back came Norwich, and Vaesen clawed away Phil Mulryne's free-kick which was bound for the top corner.
Blues defender Michael Johnson came within inches of winning it two minutes from time when he thumped a header against the post from a Grainger corner.
And so, the deadlocked teams went into a dramatic penalty shoot-out where Blues' goalkeeper Nico Vaesen saved Phil Mulryne's penalty and Daryl Sutch fired wide.
Successful spot-kicks from Stern John, Paul Devlin and Stan Laziridis set Carter up to strike the decisive penalty.
PENALTY SHOOT-OUT
NORWICH GOAL! Roberts makes it 1-0 to Norwich
BLUES GOAL! John levels it at 1-1
NORWICH MISS! Vaesen saves Mulryne kick. 1-1
BLUES GOAL! Devlin puts Blues 2-1 up
NORWICH MISS! Sutch shoots wide. 2-1 Blues
BLUES GOAL! Lazaridis makes it 3-1 Blues
NORWICH GOAL! Easton pulls it back to 3-2
BLUES GOAL! Carter puts Blues in Premiership. 4-2 Blues