The home side made a bright start and came within inches of the breakthrough on three minutes as the ball was worked to Charlie Austin, but his fierce volley struck the top of the crossbar.
At the other end, Tyrone Barnett was denied by a stunning stop by Grant before Shaun Brisley's header was cleared away from danger.
However, within 60 seconds, the Clarets were in front as a left-wing cross was headed home from close-range by midfielder Chris MCCANN who peeled off his marker to send a downward header past Olejnik.
Austin headed straight at Olejnik on 14 minutes and then Paterson drove wide from distance as Burnley came close to a second. Olejnik denied Marney on 19 minutes as the shots continued to rein on the Posh goal.
Against the run of play, Posh levelled as Shackell upended Mendez-Laing in the box and TOMLIN coolly slotted home from the spot. Incredibly on 23 minutes, Posh found themselves ahead as Tomlin found MENDEZ-LAING who coolly slotted home with his left-foot.
Mendez-Laing came close to bundling a Tomlin cross over the line with Mills doing well to clear before Brisley headed a McCann corner wide of the target.
It was 2-2 on 38 minutes as Charlie AUSTIN burst into the box and his low shot was not cleared on the line by a combination of Bostwick and Brisley.
Referee Mike Dean brought an end to the first half proceedings after a minute of stoppage time. Defensively crazy, the first 45 minutes was a case of who could take advantage of gaps at the back and with the score at 2-2, it was set up nicely for the second period.
An excellent cross from Joseph Mills picked out Austin moments after the break, but he headed wide from four yards.
Olejnik made a very good stop with his feet to deny Paterson after a clearance was charged down and in one moment, two Burnley strikers were baring down on goal. The goalkeeper stood big and made a good stop.
Lee Grant made a good stop at the other end to deny Grant McCann's left-foot shot after good work from Mendez-Laing. Gabriel Zakuani replaced Brisley on 59 minutes with Bostwick continuing to play at centre back.
A mazy run from Mendez-Laing saw him beat three defenders, but his eventual shot was deflected wide for a corner kick. Burnley then made a double switch with Sam Vokes and Cameron Stewart introduced into the fray.
Emile Sinclair replaced Boyd on 69 minutes in another very attacking change by Darren Ferguson. It was 3-2 to the Clarets on 74 minutes as more poor defending allowed Charlie AUSTIN to let fly from the edge of get box and he found the top corner expertly.
Bostwick fired wide from distance after a well worked free-kick as Posh came very close to a leveller. It was game over on 80 minutes though as AUSTIN completed his hat-trick from the spot after a foul by Olejnik on the same player. Olejnik was cautioned by referee Mike Dean.
It was 5-2 to the rampant hosts on 85 minutes as Stanislas tapped home from close-range after Olejnik had saved the initial effort from Austin.
Final Score: Burnley 5-2 Posh
Burnley: Grant, Trippier, Shackell, Duff, Mills, Marney, McCann, Wallace (sub Stewart 63min), Stanislas, Patterson (sub Vokes 63min), Austin (sub Stock 86min). Unused subs: Jenson, Edgar, Mee, Bartley.
Posh: Olejnik, Alcock, Brisley (sub Zakuani 59min), Bostwick, Tomlin, Barnett, Boyd (sub Sinclair 69min), McCann, Little (sub Ferdinand 81min), Frecklington, Mendez-Laing. Unused subs: Day, Newell, Knight-Percival, Clarke-Harris.
Attendance: 10,979 (353 Posh fans)