Ashley Barnes glanced them into a 16th-minute lead when being picked out by Elliott Bennett's free-kick from the left - and the goalscorer continued to be a pivotal figure. He was involved in a tangle on the deck with George Boyd that led to the Posh man being shown a red card.
Brighton's Bennett was scythed down by Ryan Bennett three minutes later, but the same Seagulls man saw his penalty saved by Joe Lewis - the best in a bunch of fine stops from the Posh keeper.
Brighton soon doubled their advantage in the 33rd minute when Inigo Calderon was able to convert a Bennett corner from point-blank range at the second attempt to strike for the third time in as many matches.
Livewire wideman Bennett earned another spot-kick when hauled down by Lewis in the 62nd minute but handed over responsibilities to Barnes, who stroked the ball into the corner. But the Albion frontman was left with his head in his hands moments later when he was twice denied a hat-trick. His first effort was blocked by defender Kelvin Langmead before he blazed the rebound against the bar.
And Barnes again missed out on a treble when striking the bar for a second time in the dying moments.