Striker Beckett had not started a game since November and seemed to be on his way out of the Galpharm Stadium as boss Peter Jackson makes changes in the transfer window.
While Pawel Abbott, Martin McIntosh and Danny Adams have all been told they have no future at the club and can move on, Beckett has grabbed a lifeline.
In a game spoiled by a swirling wind Huddersfield were always in control and the Robins, with a terrible away record, never looked like getting anything after Beckett struck the first goal.
Beckett claimed a typical poacher's goal when he swept home from close range after Andy Booth had headed down Gary Taylor-Fletcher's long thrown-in. That came after 31 minutes and once Huddersfield had broken the deadlock, Cheltenham's resistance was blown away.
Early on Taylor-Fletcher had a mazy run, beating three men, but his shot was turned away by Shane Higgs. Booth had two chances, first a scuffed finish which went just over the top and then he steered a header wide.
Just after Beckett's opener, Booth - looking fitter and sharper than he has for years - just failed to hit the target with a far post header which went into the side netting.
Higgs then blocked Taylor-Fletcher's powerful effort but there was no denying Booth two minutes before the break.
Cheltenham failed to clear their lines and Michael Townsend's half-clearance came to Booth just inside the box. The striker swivelled and unleashed an unstoppable drive past Higgs and into the far corner.
The game was over as a contest then and Cheltenham, having lost six of their last seven away games, hardly mustered a serious threat to the home side.
Taylor-Fletcher took advantage of a ball which stopped in the wind to rob Keith Lowe and set up Beckett but the striker was off target.
Matty Young smashed a shot wide before Chris Brandon wriggled free of his marker only to fire straight at Higgs.
Brandon had the best chance of the second half when his snap shot rattled the frame of the goal with 16 minutes left.
Somehow referee Mike Thorpe allowed more than five minutes of stoppage time, but that wasn't enough for Cheltenham and Huddersfield held out for a precious clean sheet.