The hosts were on the back foot for the majority of the first half and only two fine saves from goalkeeper Julian Speroni from Tyrone Barnett kept the scores level.
But Palace were a different proposition after the interval and Australia international Jedinak took advantage of a fumble from Posh goalkeeper Paul Jones to fire home 14 minutes from time.
Posh, who suffered their ninth defeat away from home, faded badly after making a bright start and a run of just one win in eight matches has left them looking over their shoulders.
Palace, by contrast, extended their unbeaten run to six matches in the npower Championship and they are now eight points behind the play-off places.
Dougie Freedman made just one change to his Palace side from their goalless draw at West Ham last week, with Kagisho Dikgacoi making way for Owen Garvan in midfield.
Craig Alcock returned from a hamstring injury to replace Ryan Bennett at the back in one of four changes to Posh's line-up from their 1-1 draw at Doncaster.
The visitors came firing out of the traps and Tyrone Barnett's strike from just outside the area forced Speroni to tip over.
Palace's goalkeeper again magnificently thwarted Barnett after the striker had cut through the defence.
Despite their dominance in the early stages, Posh were nearly caught out but Darren Ambrose could only fire high and wide following good work by Wilfried Zaha.
Tommy Rowe headed over the bar as Peterborough continued to make a mockery of their record away from home.
Palace did start to get to grips with their challenge after Ambrose's miss and it was the winger who went closest next, with his effort from Jermaine Easter's lay-off deflected onto the upright by Gabriel Zakuani.
Jedinak was then felled by a challenge in the area on the stroke of half-time but appeals for a penalty were waved away by referee Scott Mathieson.
Patrick McCarthy headed wide from Ambrose's free-kick as Palace picked up where they left off after the interval.
The defender should perhaps have directed his header on-target after putting himself in a good position, while Chris Martin also missed the target from the edge of the area after being teed up by the influential Zaha.
Midfielder Zaha then narrowly shot wide from just outside the box after picking up Ambrose's ball, while at the other end Rowe went close with his effort from 20 yards.
Palace grew in stature thereafter and Jedinak's downward header produced a fine block from Jones.
But Posh's shot-stopper made a glaring error soon after as Jedinak put the hosts in front in the 76th minute.
Ambrose's floated free-kick from the left was fumbled by Jones to the feet of Jedinak, who swivelled and fired into the net.
The 27-year-old then put Glenn Murray through on goal but Jones did well to deny the substitute's drive.
Jones also spread himself to stop Dean Moxey mere minutes after the defender had come onto the pitch to keep the deficit to just one goal.