Beckford, on loan from Bolton, collected a pass from Kevin Davies, breezed between two defenders and coolly clipped the ball past goalkeeper Ben Alnwick in first-half stoppage time.
And that proved enough to ensure Simon Grayson’s 100th match at the North End helm was a winning one as free-falling Posh slumped to a fourth successive home league defeat.
Grayson, who took charge of Preston in January 2013, had earlier seen Paul Gallagher’s early shot crash back off the crossbar before defender Paul Huntington steered a free header from a Gallagher corner over the crossbar.
Peterborough offered little after Erhun Oztumer had seen a second-minute free-kick saved. North End keeper Thorsten Stuckmann blocked bravely after a Huntington error had let in Aaron Mclean early in the second half and Mclean was off target with the hosts’ best chance of all when slamming strike-partner Craig Mackail-Smith’s cross wide with 15 minutes to go.