Sinclair, a deadline day signing from Macclesfield, opened his Peterborough account with three minutes on the clock and capitalised on a mistake from Burnley centre-back David Edgar to double his tally seven minutes before half-time.
Keith Treacy reduced the arrears with a sensational strike on the stroke of the interval but, despite contributing fully to an open game, Burnley's final ball let them down on numerous occasions.
And when they did find a way through, Peterborough goalkeeper Paul Jones was in inspired form.
As a result the hosts were able to hold out for a first victory in four games since dismantling Ipswich 7-1 last month.
Eddie Howe's Burnley are now without a clean sheet in 20 matches and have a solitary win to show for their endeavours this term.
Peterborough made the perfect start when Sinclair collected Lee Tomlin's pass to find the bottom corner from the edge of the area.
The home side were rampant in the opening exchanges and defenders Ryan Bennett and Ryan Tunnicliffe both headed off target from Grant McCann corners.
When retrieving the ball after the latter set play in the 14th minute, Burnley keeper Brian Jensen slipped and suffered an apparent groin injury that ended his involvement.
Despite that blow the Clarets started to feel their way back into the contest, forcing Jones to make a quartet
of impressive saves in quick succession from Charlie Austin, Jay Rodriguez, Chris McCann and Edgar as winger Ross Wallace came to the fore in much of the visitors' work.
Burnley were controlling possession in attractive fashion as half wore on only for an individual error to undermine their efforts in the 38th minute.
Edgar clumsily surrendered possession to Sinclair who was able to carry the ball into the box and fire past substitute goalkeeper Lee Grant.
Treacy capped an entertaining half with a magnificent strike that dragged Burnley back into the contest. The former Blackburn winger cut in from the left and unleashed a right-footed piledriver from 25 yards into the top corner.
Sinclair had the chance to complete his hat-trick shortly before the hour when Tomlin and McCann combined down the Burnley right, but he directed his shot into Grant's midriff.
The Peterborough forward spurned a clearer opportunity moments later when he touched Bennett's long ball away from Edgar and raced towards the Burnley box, only to fire wide on the half volley.
The visitors felt they might have had a penalty after Rodriguez chased down Treacy's pass to get goalside of Craig Alcock before taking a tumble in the box. Referee Jock Waugh was uninterested.
Austin then unleashed a fierce left-footed drive, but Jones was once again equal to the task as he tipped over.