The contest came to life in the 13th minute when Posh had the ball in the back of the Albion net only for their celebrations to be cut short by referee Steve Bratt.
Wales international defender Craig Morgan rose to meet a left-wing corner from Russell Martin and steered a powerful header past the motionless Michels Kuipers, but Bratt judged him guilty of pushing defender Matt Richards and immediately scrubbed off the goal.
Morgan was close with another header from a curling Tom Williams cross moments later, before the focus of attacking attention switched to the other end where Albion striker Glenn Murray was found badly wanting.
The former Rochdale striker was picked out by Joe Anyinsah's low delivery but blazed a first-time effort high and wide.
Posh produced most of the attacking spark after the break but Albion defended in depth with Tommy Elphick and Colin Hawkins giving top home marksman Craig Mackail-Smith little scope to add to his ten-goal tally.
Referee Bratt then found himself centre-stage and after yellow-carding three players from either side, the West Midlands man had to stand down with a muscle injury in the 73rd minute and was replaced by a fourth official Paul Barnes.
Brighton's first shot on target did not come until five minutes later when Kevin McLeod brought the best from Posh's Joe Lewis, although it was the home side who finished strongly with Brighton under pressure for almost eight minutes of added time.
Charlie Lee was closest to breaking the deadlock with a powerful header before substitute Dominic Green blazed high over the bar.