Buckley, who struck Albion’s second goal in a 2-0 win against Bristol City in the league last weekend, netted both his team’s goals at London Road.
He took the credit for the first one but it came about courtesy of a mistake by Posh goalkeeper Joe Lewis.
The second was a clinical finish from the edge of the box to put Brighton back in front late on, after substitute David Ball had equalised for the hosts.
Tommy Rowe returned from an Achilles injury as one of three changes made by Peterborough boss Darren Ferguson.
Rowe was recalled in midfield along with Scott Wootton as Joe Newell and Daniel Kearns dropped out of the starting XI. Paul Taylor replaced George Boyd in attack.
Posh skipper Grant McCann was included on the bench after a five-week lay-off with knee ligament damage.
Changes were probably inevitable for Brighton after they were taken to extra time and penalties by Wrexham in the FA Cup in midweek, and boss Gus Poyet made four.
Captain Gordon Greer returned after suspension while fellow defender Marcos Painter and forward Jake Forster-Caskey were back after injury. Midweek goalscorer Ashley Barnes was rewarded with a start, replacing Will Hoskins up front.
However, Painter lasted just six minutes on his return before he was forced off injured. Adam El-Abd came on in his place.
El-Abd’s first contribution was to foul Lee Tomlin. From the resulting free-kick, Ryan Bennett headed Wootton’s cross narrowly wide.
Matt Sparrow tried his luck from outside the area either side of the midway point of the first half but his shots went wide on each occasion, before a third attempt forced Lewis to parry the ball away.
Craig Mackail-Smith curled an effort wide against his former club after half an hour.
Two minutes later Brighton were ahead after Sinclair fouled El-Abd wide on the left. Buckley took the resulting free-kick and his ball into the box floated over Lewis’ head and into the net.
Peterborough responded well and Peter Brezovan had to get down to deny Sinclair before Ryan Tunnicliffe sent a shot over the crossbar.
Sparrow threatened again early in the second half but Posh were soon on the offensive again and Taylor tested Brezovan with a 20-yarder.
Ball levelled after 72 minutes, scoring from close range after being teed up by Tunnicliffe.
It looked like that would be enough to earn Ferguson’s side a point as the game seemed to peter out afterwards.
However, with two minutes remaining Buckley beat Lewis with a drive from the edge of the box that flew into the roof of the net.