Dennis Wise's side dominated much of the game but they failed to carve out many clear-cut chances and were even forced to repel some second-half pressure from the Cheshire side.
But it was Beckford's finish from debutant Peter Sweeney's hanging cross that proved decisive and moved Leeds up to third in the League One table.
United skipper Andrew Hughes posted the first threats with a thumping volley which flew over and a header which he glanced from Sweeney's corner onto the woodwork.
Crewe managed to break out of their own half after 19 minutes and Leeds were indebted to keeper Casper Ankergren for keeping Michael O'Connor's downward header from creeping in at the near post.
Once Wise's side were in front they exerted a steely grip on proceedings, retaining large amounts of possession and forcing Crewe's centre-halves, Julien Baudet and Chris McCready, to work overtime keeping veteran frontman Tore Andre Flo and Beckford in tow.
That they did was a credit to their application as Leeds were restricted to shooting from outside the box. One such effort from Frazer Richardson at the start of the second half should have produced a penalty for the visitors as it clearly struck Ben Rix's hand.
Within a minute Rix could have put Crewe level when Ryan Lowe played him through, but Ankergren came off his line smartly to block the midfielder's effort.
Crewe had their own appeals for a spot-kick waved away just after the hour mark when Nicky Maynard appeared to be held back as he latched on to a Neil Cox header.
Byron Moore injected some pace on the right when he came on as Crewe went in search of an equalizer and it almost came when McCready crashed a header onto the bar from a corner.
But Leeds substitute Tresor Kandol could, on several occasions with more thought, have played Beckford in to finish the game off in the closing minutes.
Crewe's hopes of late salvation were dashed when centre-half Baudet limped off with cramp after Alex's first-team coach Steve Holland had used all three substitutes.