With Falkirk looking the more likely to snatch a win in an untidy game, the Lithuanian striker made the difference.
Hearts looked lively early on and Mark Howard saved well from Rudi Skacel after only eight minutes.
The Edinburgh side then had a claim for a penalty turned down by referee Alan Freeland after Patrick Cregg challenged Jankauskas from behind in the area.
But Falkirk weathered the early storm, and Tiago Rodrigues should have opened the scoring after 21 minutes when he had a free header in front of goal but glanced his effort wide.
It proved a costly miss as Hearts equalised within a minute. Skacel played over a low cross from the left, and Paul Hartley rushed in to bundle the ball past Howard from close in.
Falkirk equalised on the stroke of half-time when Alan Gow's 20-yard free-kick struck a hand in the Hearts' wall and this time the referee pointed to the spot. Craig Gordon saved Gow's initial spot-kick, but the striker followed up and lashed home the rebound.
The game threatened to boil over in the second half with five bookings in the space of 10 minutes as Mr Freeland struggled to keep control.
But there was little in the way of goalmouth incident, and Falkirk changed things by bringing on Russell Latapy for the final 20 minutes.
But it was Hearts who snatched the win when Rodrigues needlessly gave away possession and Calum Elliot picked out Jankauskas, whose deflected shot beat Howard.
A goal from Edgaras Jankauskas with only nine minutes left gave troubled Hearts a valuable three points.
With Falkirk looking the more likely to snatch it in an untidy game, the Lithuanian striker made the difference.
Hearts looked lively early on and Mark Howard saved well from Rudi Skacel after only eight minutes.
The Edinburgh side then had a claim for a penalty turned down by referee Alan Freeland after Patrick Cregg challenged Jankauskas from behind in the area.
But Falkirk weathered the early storm and Tiago Rodrigues should have opened the scoring after 21 minutes when he had a free header in front of goal but glanced his effort wide.
It proved a costly miss as Hearts equalised within a minute. Skacel played over a low cross from the left and Paul Hartley rushed in to bundle the ball past Howard from close in.
Falkirk equalised on the stroke of half-time when Alan Gow's 20-yard free-kick struck a hand in the Hearts' wall and this time the referee pointed to the spot.
Craig Gordon saved Gow's initial spot-kick but the striker followed up and lash home the rebound.
The game threatened to boil over in the second half with five bookings in the space of ten minutes as Mr Freeland struggled to keep control.
But there was little in the way of goalmouth incident and Falkirk changed things by bringing on Russell Latapy for the final 20 minutes.
But it was Hearts who snatched the win when Tiago Rodrigues needlessly gave away possession and Calum Elliot picked out Jankauskas whose deflected shot beat Howard.