The striker opened the scoring on nine minutes, turning onto Michael Doyle's pass to shoot past Mikkel Andersen. Sam Vokes added a second with a close-range shot from a 55th-minute cross by Robert Snodgrass.
The unstoppable Beckford wrong-footed Byron Anthony to make it 3-0 with an angled drive on 64 minutes and substitute Tresor Kandol completed the rout with a tap-in three minutes from time.
United were simply in a different class to their hapless opponents who have now lost four games in succession after a bright start to the season.
The closest Rovers came was a 21st-minute volley from skipper Stuart Campbell which Leeds' goalkeeper Casper Ankergren kicked onto his crossbar. The ball bounced down onto the goal line, but did not cross it.
Leeds had plenty of efforts at goal and kept Andersen busy during a one-sided contest which had their 1,600 travelling fans celebrating long before the final whistle.