The league leaders looked set for a routine afternoon when Genero Zeefuik netted early on, but they fell behind to a John Baird spot-kick and a Rory Loy effort.
Although the hosts levelled through substitute James Keatings, in their pursuit of a winner they left the back door open as Sibbald’s late strike upset the odds.
The game was just a minute old when Zeefuik opened the scoring – capitalising on a surging run down the left by Osman Sow to tuck home from close range and ignore the home crowd.
An imaginative back-heel from Sow nearly doubled the lead moments later but Falkirk served a warning of their intent when a free-kick caused confusion in the home defence and Thomas Grant sent a great chance over the crossbar.
Sam Nicholson then dragged an inviting opening wide of the target and the end-to-end nature of the game continued with home goalkeeper Neil Alexander doing brilliantly to prevent Luke Leahy from equalising.
With half an hour gone, Falkirk had their leveller when Jordan McGhee clumsily felled Loy and John Baird stepped up to coolly steer the resulting penalty into the top corner of the net and Hearts could have trailed at the break had Alexander not kept out a fierce Blair Alston long-ranger.
With seven minutes of the second half gone, Tynecastle was silenced again – Loy sending Falkirk into an unlikely lead with a superb curling shot that found the top corner.
Hearts responded quickly and could have been level had Prince Buaben placed his shot beyond the reach of Jamie MacDonald, and the visiting goalkeeper was alert again on the hour mark to parry a point-blank header from Nicholson.
The hosts threw on Keatings as they looked to restore parity and the striker had already passed up one golden opportunity by the time he smashed home a loose ball from close range with 17 minutes of normal time remaining.
Tynecastle breathed a sigh of relief and many will have expected them to capitalise on the momentum and preserve their unbeaten record with victory, but with 10 minutes remaining they were left stunned when Sibbald pounced on a sloppy clearance to rifle the winner past Alexander.