Max Gradel also netted for Leeds after Carlisle had gone in level at half-time against the run of play thanks to a Richard Keogh header.
Leeds set the early pace and it was no great surprise when Argentine striker Becchio put them in front in the 13th minute, applying the vital touch from close range after Carlisle keeper Adam Collin had spilled Robert Snodgrass' free-kick.
Carlisle grabbed the equaliser in the 43rd minute after Leeds defender Richard Naylor fouled Gary Madine 35-yards from goal. Ex-Leeds stalwart Ian Harte curled in the free-kick with pace and defender Keogh was on the end of it to thump a header into the bottom corner from six yards for his fifth goal of the season.
There was no respite for Carlisle after half-time and Gradel restored Leeds' advantage in the 47th minute, heading Leigh Bromby's right wing cross home from six yards.
The visitors went 3-1 up four minutes later when Becchio left Tom Aldred in his wake as he raced away from just inside the Carlisle half and then cut into the penalty area to coolly slide his shot past Collin for his 15th goal of the season.