The visitors were the stronger side and although Carlisle enjoyed plenty of possession in spells either side of half-time Leeds soaked up the pressure well and were always a big threat on the break.
Leeds, unbeaten away from home, took their time to settle, but once their midfield asserted itself it was Carlisle's turn to defend as Jonathan Douglas pulled a shot narrowly wide after working a smart move with ex-Carlisle loanee Beckford.
The Yorkshiremen grabbed a 31st-minute lead when Neil Kilkenny's corner from the left flew across the face of goal and Paul Telfer poked the ball back into the danger area where Argentinian striker Becchio reacted quickly, swooping to force the ball home from six yards.
Carlisle's top-scorer Danny Graham didn't reappear for the second half after suffering a bang on the head just before half-time but substitute Scott Dobie went agonisingly close with a header which flew inches too high.
Leeds made the game safe four minutes from time when substitute Robert Snodgrass got away on the left and crossed a perfectly-weighted low ball which enabled Beckford to slide home from six yards.