An opportunity fell to Andrew Crofts at the other end moments later, only for the midfielder to sky his effort.
Gus Poyet's men looked lively in the second period, and had the chance of the game on 46 minutes when William Buckley forced a magnificent save from Olejnik.
They were denied by the woodwork moments later when Andrea Stabilin Orlandi saw his stinging strike come crashing back off the post.
The Seagulls continued to bang on Peterborough's door. Wayne Bridge was next to test Olejnik and the Austrian shot-stopper tipped his 78th-minute effort over the crossbar.
Brighton got a deserved result at the death through Dobbie. The Scottish striker pounced in the box in the second minute of stoppage time to snatch the points and break Posh hearts.