The Czech has struggled with injuries but his contribution was vital with Falkirk looking odds on to hold out for a point.
The match was played in shocking conditions with strong winds, sleet and snow all making good football virtually impossible.
So both teams deserve credit for serving up good entertainment in the circumstances.
Kasper Schmeichel showed he is a chip off the old block as he kept Hearts at bay with his string of first-half saves.
He got down to his right to push Paul Hartley's 25-yarder round the post and then used his legs to block Andrius Velicka's angled drive.
But Falkirk had chances too and Steve Banks, deputizing for the injured Craig Gordon, made a fine stop from Jack Ross although Darren Barr should have scored from the rebound which struck the post.
In the second period, Hartley was close with a 20-yard free-kick and Velicka almost bundled a cross from Hartley over the line with his body, but was denied by the upright.
Similarly Vitor Lima cracked the post for Falkirk when he ran on to Thomas Scobbie's cross.
But the points were settled in 74 minutes when Bednar reacted quickest to Andrew Driver's cross from the right, to fire a low finish past Schmeichel's despairing dive.
The game ended in controversy when Barr was ordered off for supposedly head-butting Velicka.
However, TV evidence suggests that there was little contact and the Hearts player should be ashamed of his reaction.