United's win made it 10 successive home league wins for the first time in 40 years - and put a dent in Tranmere's own play-off ambitions.
Beckford returned from injury to net his 32nd goal of the season after 15 minutes, delivering a cool finish after trading passes with Robert Snodgrass in a sweeping move started by Andy Robinson deep inside the former Tranmere winger's own half.
Kilkenny, looking suspiciously offside, latched on to a long pass from Jonathan Douglas to fire home a low shot from 15 yards for the second but a minute later Leeds defender Sam Sodje headed over keeper Casper Ankergren for an embarrassing own goal.
Anthony Kay and Edrissa Sonko both went close for Tranmere and Ankergren made a stunning double save from Kay and Ryan Shotton.
But Leeds scored a third shortly after half-time when Beckford helped on a free-kick from Snodgrass and the ball went in off Becchio's shin.