Buckley won the game for Albion shortly after they, like Leicester, had been reduced to 10 men.
The Foxes went on to finish the match with nine players.
It was an entertaining encounter between two evenly-matched sides but chances were few and far between.
It sprung into life when Beckford was controversially sent off in the 53rd minute.
Referee Andy Woolmer evened things up numerically with five minutes remaining by showing Matt Sparrow a second yellow card.
Brighton were not deterred and took the lead through Buckley, and it might have been 2-0 in the end had Neil Danns not also been shown a red card for hauling down substitute Rodriguez Vicente in injury time.
Brighton boss Gus Poyet, who this week ruled himself out of the running for the vacant managerial position at Leeds, named an unchanged side.
Former Foxes full-back Joe Mattock was on the bench after joining Albion on loan from Portsmouth in a deadline-day deal.
Danny Drinkwater was handed his full debut in the only change made by Leicester. The former Manchester United trainee replaced Paul Gallacher who dropped down to the bench. New signing Ben Marshall was also among the substitutes for Nigel Pearson's side.
Leicester were presented with a good opportunity after 13 minutes when Adam El-Abd fouled Beckford just outside the area, but Paul Konchesky's curling free-kick was tame and easily saved by Peter Brezovan.
Ashley Barnes headed wide for Brighton before David Nugent did likewise for Leicester either side of the half-hour mark.
Beckford spurned the chance of the game just before half-time.
The Brighton defence were caught out by a long ball over the top and Beckford, scorer of six goals in his last four matches prior to today, found himself clean through on goal. But his side-foot finish was weak and Brezovan made the block.
Beckford was in the thick of the action again soon after the break.
After mistiming a challenge on Lewis Dunk down near the corner flag, Beckford tapped the Brighton defender on the back and exchanged words with him.
This led to the referee's assistant flagging furiously and Beckford being shown a straight red card by referee Andy Woolmer, much to the disbelief of the Leicester players.
Brighton should have made their numerical advantage count in the 68th minute when Buckley went through one-on-one but Kasper Schmeichel got his angles right to block his shot.
Sparrow was dismissed in the 85th minute after a reckless lunge on Leicester substitute Marshall.
It looked like heading for a stalemate but in the final minute Vicente's cross from the right eluded the whole Foxes backline and Buckley volleyed home at the far post.
Leicester's misery was compounded in injury time when Danns hauled down Vincente when he was bearing down on goal and was sent off.