Beckett, who was recalled to the starting line-up by manager Andy Ritchie, finished with aplomb with 12 minutes remaining, after Andrew Whing was adjudged to have tripped Chris Brandon in the penalty area.
Huddersfield seized the initiative early on and it was no surprise when they took the lead with four minutes on the clock, albeit from an unlikely source.
Full-back Andy Holdsworth, who had not found the net for over ten months, exchanged passes with Beckett on the edge of the D before rifling a shot past Michels Kuipers.
The visitors could have equalised ten minutes later through Nicky Forster, but Brighton's top scorer flashed his shot wide.
A superb move involving Frank Sinclair and Beckett allow Michael Collins a sight at goal and he forced Kuipers into a smart stop.
Andy Booth nearly made it 2-0 not long after from a Holdsworth corner, but Kerry Mayo cleared his goalbound header off the line.
Dean Wilkins' half-time pep talk worked the trick for Brighton, with the Seagulls coming out after the interval with renewed vigour.
Their dangerman Glenn Murray was awarded a penalty within a minute after Holdsworth was penalised for bringing the striker down. Forster stepped up to level the scores.
Only a wonderful reflex save from Terriers goalkeeper Matt Glennon prevented Joel Lynch from putting Brighton ahead on 52 minutes.
A dipping shot from Murray was then miraculously palmed over the top by Glennon, as Brighton pressed forward in numbers looking for a second goal.
And although Beckett's penalty turned out to be the winner, Brighton went close to equalising in the dying seconds through Murray, whose header ricocheted off the woodwork.