The young midfielder hauled the Yorkshiremen back into the tie early in the first period at Brunton Park and with the clock ticking down towards extra-time arrowed the winner into the bottom corner from 15 yards.
Howson's double completed a surprise turnaround for Gary McAllister's side, who had been outplayed in the first leg at Elland Road to trail 2-1 on aggregate ahead of Thursday night's return.
Most in a crowd of 12,872 were expecting John Ward's side, playing Conference football three years ago, to finish the job.
But Leeds now head back for their second play-off final in two years and a chance to erase the memory of their defeat to Watford at the Millennium Stadium in 2006.
The tie hung in the balance as referee Alan Wiley waved away Carlisle's early shouts for a penalty when Scott Dobie went tumbling in the penalty area after Carlisle's Grant Smith had misdirected a long-range shot.
But Leeds were on target at the end of their first attack to level the tie in the 10th minute.
Howson sent Dougie Freedman scampering down the left and raced into the area to latch on to the on-loan Crystal Palace striker's cross and fire home from close range.
A neat interchange between Neil Kilkenny and Jermaine Beckford saw the former's volley comfortably held by home goalkeeper Keiren Westwood.
In contrast to the first leg at Elland Road it was Leeds who had settled quickest into their stride.
Evan Horwood almost gifted Leeds a second 10 minutes before the break when his weak back pass was seized on by Freedman, but the forward's shot was snaffled by Westwood.
At the other end Dobie miscued his shot and it was clear Carlisle had lost some of Monday night's zip and accuracy.
But right at the end of the first half, Marc Bridge-Wilkinson went close to firing the Cumbrians ahead on aggregate when his firm low drive was smothered by Casper Ankergren.
Bridge-Wilkinson gave the ball away shortly after the restart and Freedman pounced again to test Westwood from 25 yards.
Kilkenny then dragged a right-footed shot wide when teed up by Beckford, but for a 15-minute period in the second half neither side carved out an opening.
Bridge-Wilkinson dithered on the edge of the area when well-placed following Simon Hackney's cross and allowed Bradley Johnson to block his shot as the tension mounted.
Kilkenny, Howson and Prutton combined in the box, but the latter's shot was blocked, while Bridge-Wilkinson flashed a volley just wide.
Leeds then began to pen in their hosts for long periods. Westwood flung himself low to his right to turn away Freedman's 25-yard shot and Johnson's thumping header hit the outside of the post.
Leeds looked the more composed as full-time beckoned and Jonathan Douglas' deflected shot arced over the crossbar as the tie headed towards extra time.
Then with one minute of time added on to play Howson snatched victory for Gary McAllister's side.
The ball broke to the young midfielder on the edge of the area and the 20-year-old drilled a slightly scuffed effort into Westwood's bottom left-hand corner.