The match was evenly poised in the first half with the score 2-2 at the interval, but Burnley fired home three unanswered goals in the second period to secure their second league win of the season and leave Peterborough still waiting for their first.
Burnley took the lead on seven minutes as Chris McCann's header from six yards beat Peterborough keeper Robert Olejnik.
Posh responded to going behind by launching several attacks on Burnley's goal with Grant McCann's strike from 18 yards out blocked by goalscorer Chris McCann in the eighth minute, before Lee Grant saved George Boyd's long-range effort four minutes later.
Peterborough's pressure paid off in the 19th minute when they were awarded a penalty. Lee Tomlin calmly slotted home, beating Grant with a shot into the bottom left side of the goal.
Four minutes later, Peterborough took the lead when Nathaniel Mendez-Laing's left-footed effort beat Grant from 12 yards out after Tomlin headed the ball across from the right channel.
Seven minutes before the break an Michael Bostwick own goal enabled Burnley to draw level after the midfielder put the ball into his own net from six yards out.
Both sides remained unchanged at the interval and they started the second half brightly with several chances apiece in the first 10 minutes after the re-start, although none of them resulted in a goal.
Burnley defender Joseph Mills was booked for unsporting behaviour in the 54th minute, while Peterborough manager Darren Ferguson was forced to make the first substitution of the game with Gabriel Zakuani replacing the injured Shaun Brisley.
Burnley manager Eddie Howe responded with two substitutions of his own as Ross Wallace was replaced by Cameron Stewart and Sam Vokes came on for Martin Paterson.
The home side had a couple of chances courtesy of Junior Stanislas and Stewart before Ferguson made his second substitution as Emile Sinclair came on for George Boyd in the 69th minute.
Burnley's dominance paid off in the 74th minute when Austin gave his side a 3-2 lead from 18 yards out after fine work by Chris McCann down the left flank.
The Clarets put the result beyond doubt when Austin scored his second goal of the game from the penalty spot after he was fouled by Olejnik to make the score 4-2 10 minutes from the final whistle.
The icing on the cake for Burnley came five minutes from time when Stanislas scored from 12 yards out after being fed from the left channel by Mills.