Conor Washington and Michael Bostwick scored in the last three minutes to condemn Crawley to a defeat that leaves them stuck in the relegation zone going into the final week of the season.
Saunders’ side had twice hit the front at the ABAX Stadium before being undone in the dying moments.
Marvin Elliott hit their opener just before the half-hour mark only for Washington and Bostwick to strike either side of the interval to edge Posh ahead.
Crawley responded well with Mathias Pogba levelling before former Peterborough midfielder Josh Simpson struck for Crawley midway through the second period.
Saunders’ side held that advantage until Bostwick levelled and Washington struck the decisive blows at the death.
Saunders, whose team face fellow strugglers Coventry in their final match, said: “It’s just unbelievable as we had it won with three minutes to go.
“That’s football though, isn’t it? They did not look like scoring and I thought we were in control, but then they came up with two goals.
“The lads are devastated in the dressing room. When we needed a bit of luck we didn’t get it.
“I would have been disappointed with a point in all honesty, yet we have now come away with nothing.
“We are still alive though and it looks like we have to win next week to stay up. We just need to play with belief.”
Peterborough handed a debut to 16 year-old schoolboy Leo Da Silva Lopes with the play-offs now beyond them.
He was taken off seconds before Simpson fired Crawley ahead, but those late goals ensured he contributed to a winning performance.
Bostwick blasted in from the edge of the area in the 87th minute before Washington pounced after skipper Gaby Zakuani had headed against a post.
Interim Peterborough boss Dave Robertson said: “We wanted to go out with a bang at home and we have done that.
“To win a game like that with two under 18s and a 16-year-old, who is going to back to do his GCSEs, is fantastic.
“I think Leo did well. He grew into the game and he understands he needed to play a little bit quicker. They got after him early on.
“The decision about who will manage next season will be made over the next couple of weeks, but I want to continue to work with this group of players and develop them.”