The 17-year-old academy product, who agreed fresh terms with Leeds on Friday, produced his first senior goal with a brilliant 30-yard strike to set up United's first League One victory since their visit to Scunthorpe United on the opening day of the season.
Delph found the net with a long-range shot on 27 minutes which beat goalkeeper Steve Collis with the aid of a post, and Alan Sheehan doubled Leeds' advantage nine minutes later when his fierce shot from 25 yards found the net with a flick off the crossbar.
Crewe created few chances of note, with Byron Moore's dipping volley on 11 minutes their best effort of the match.
But Douglas fired home another long-range strike four minutes into the second half to put the outcome beyond doubt, and Beckford's eighth goal of the season on 66 minutes was followed by Andy Robinson's late curling effort which rounded of the afternoon in style.
There was bizarre drama to come, however, as Lubomir Michalik was sent off in injury-time before Crewe earned two rapid but meaningless consolations through Billy Jones' free-kick and Eugene Bopp's deflected finish.