Beckford, returning after injury, scored in the 62nd and 71st minutes to take his season's tally to 21 and Howson added Leeds' third in injury-time.
Craig Mackail-Smith netted his 19th goal of the season with 10 minutes left but it wasn't enough as Leeds turned on the style.
Whites goalkeeper Casper Ankergren and his Peterborough counterpart Joe Lewis kept the first half goalless with spectacular saves before Beckford transformed the match after leaving the bench in the 58th minute.
Firstly he slotted home the rebound after Lewis had parried Bradley Johnson's free-kick and the 25-year-old then brilliantly deceived Craig Morgan before firing the ball into the bottom-right corner.
Mackail-Smith netted for Peterborough after Boyd headed against the bar but Howson sealed Leeds' win following a pass from fellow substitute Robert Snodgrass.