But Carlisle put on an impressive performance and Leeds were given an early scare when a tame headed clearance resulted in Bridge-Wilkinson volleying against the outside of the post from a Graham cross.
Manager Gary McAllister's side responded with Freedman almost putting them ahead only for Carlisle keeper Westwood to excellently palm away his curling shot at full-stretch.
The home side looked uncertain at the back and Graham had a far post header saved before Westwood bettered his earlier save by getting his fingertips to Beckford's goalbound shot.
It proved a crucial save as Carlisle immediately went up the other end to rock their rivals by deservedly taking the lead from a corner which should have been a goal-kick.
Bridge-Wilkinson (right) scored Carlisle's second at Elland Road
The corner piece was only cleared to Hackney and he unleashed a volley from the edge of the area which took a deflection off Graham's back to leave Casper Ankergren rooted to the spot.
Westwood was playing a pivotal part in the game and he again denied Beckford by saving the striker's close range angled shot.
Carlisle showed more energy and movement than their rivals and Bridge-Wilkinson pounced on some sloppy Leeds play only for Ankergren to save his effort.
But, almost immediately, the Carlisle midfielder was on hand to sidefoot in from Evan Horwood's low cross to extend his side's lead.
Carlisle grew more intent on protecting their lead as the game progressed and Westwood again foiled Leeds as he tipped over a looping header which came off team-mate Paul Arnison.
Westwood was finally beaten when a long-diagonal free-kick deep into stoppage time fell to Freedman and the Leeds forward put a close-range shot through the keeper's legs.
Leeds boss Gary McAllister: "It (Freedman's goal) keeps us in the tie.
"The goal just puts a different picture on the game.
"I've got to applaud my players for their effort. They kept going way past the 90 minutes."
Carlisle manager John Ward: "We've come to Leeds with a 40,000 crowd baying for blood and have stood up to it exceptionally well.
"Not only that but we have played some good football. Both teams have tried to attack, go forward and provide entertainment as well as try to win.
"I've got to be happy with a goal lead as things could be a lot worse."
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