Brighton took a big step towards the play-offs with their eighth home win out of nine.
But Peterborough are still looking over their shoulders at the relegation zone after suffering only their second defeat in 10 games. A controversial penalty decided an untidy match.
Referee Iain Williamson ruled that Peterborough captain Guy Branston, on loan from Rotherham, handled a shot by John Piercy from a corner.
Leon Knight slotted his 26th goal of the season from the spot.
Peterborough boss Barry Fry protested: "I thought it was a very harsh penalty. There was no way Branston could get out of the way of that. It hit him on the chest, hit his hand and he did not know anything about it." Brighton manager Mark McGhee admitted: "I could not tell you what went on. There were a load of bodies in there.
"For about 20 seconds I did not know what the referee had given, whether it was a penalty to us or a free kick to them.
"We've had some decisions recently which we've felt were hard. We had a lot of other chances to finish the game off which we did not take, but I feel our win was justified." Former Bradford midfielder Paul Reid, making his full debut for Brighton, twice went close to stretching their lead in the second half.
Peterborough were always a threat and Ben Roberts had to save shots after the break from Matthew Gill and Andy Clarke.
Fry added: "We huffed and puffed but never really looked like scoring. It was a poor pitch and a poor game from our point of view." Brighton now need a maximum of five points from their remaining games away to Bristol City and Wrexham and at home to Notts County to be certain of a play off place.
McGhee said: "I think we will end up needing only one more win, but we cannot be complacent.
"We've had a great result but now we have a really difficult game coming up against Bristol City."