Leeds took a stranglehold on the game as the goal proved a body blow which took the wind out of Carlisle.
Beckford headed wide for the visitors from a Bradley Johnson cross and Freedman shot wide as the Leeds strikeforce looked to take advantage of their side's dominance.
Carlisle almost helped them when a tame header by Evan Horwood let Freedman in on goal but keeper Keiren Westwood saved the defender's blushes by quickly rushing out to block the striker's chip.
The home side regrouped and, on the stroke of half-time, Marc Bridge-Wilkinson struck a 12-yard shot with venom, only for Ankergren to save.
Graham sent a header straight at Ankergren as Carlisle worked another excellent chance shortly after the interval as they tested Leeds' resolve.
The visitors held out and came agonisingly close to taking the lead on the counter-attack with a piercing move.
A neat one-two between Howson and Neil Kilkenny on the edge of the area worked David Prutton in on goal and his angled shot from close range was blocked by Danny Livesey.
Leeds regained the ascendancy and Freedman's shot forced Westwood into a full-length save before Johnson headed a corner against the outside of the post.
The tension increased the longer the game went on and, just when it seemed extra-time was inevitable, Howson shot into the corner of the net to put Leeds one step away from promotion to the Championship.
Carlisle manager John Ward: "I thought in the last 20 minutes of the whole match there were two very tired teams out there.
"It's a shame we lost in terms of the season but there is a team here for the citizens of Carlisle to be proud of."
Leeds boss Gary McAllister: "I thought we deserved it tonight.
"I feel for Carlisle. They have been outstanding this season but we are the victorious ones and going to Wembley."
Leeds scored after 96 minutes in the first leg and McAllister added: "I think psychologically that helped us."
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