David Obua and Andy Driver wasted first-half chances for the Edinburgh side and Falkirk battled their way to a well-deserved point.
Kjartan Finnbogason wasted the visitors' best chance in front of goal prior to the break on a dismal day for the home side, whose captain Michael Stewart was jeered off by his own fans when he was substituted in the 79th minute.
Hearts made a good start and should have been in front after the fifth minute.
The ball made its way across to Lee Wallace on the left flank and the full-back's delivery was met by Obua in the six-yard box but he squandered a great chance as he failed to keep his header down.
Falkirk had a sight at goal in the 12th minute when a poor defensive header by Ismael Bouzid from Marc Twaddle's long ball fell at the feet of Finnbogason, but the Icelandic marksman took too long and Janos Balogh was able to come out and smother the ball.
Christian Nade then showed neat close ball control to shoot 25 yards from goal but his stinging effort flew over the bar for Hearts.
Driver wasted a chance in the 35th minute when, after exchanging passes with Suso Santana, the Englishman raced clear of the defence before shooting wide from an angle.
Just seconds later, Falkirk raced up the field and Alex MacDonald forced a save from Balogh with a long-range effort that skidded off the wet surface.
Falkirk captain Darren Barr was relieved to still be on the pitch after pulling Santana down on the edge of the box as the midfielder raced clear. Referee Chris Boyle showed the Scotland defender a yellow card.
Falkirk midfielder Scott Arfield then flashed a shot wide from 18 yards in first-half injury time.
Falkirk had Hearts under pressure 11 minutes into the second half as the hosts failed to get the ball out of their own box.
The attack ended when Ryan Flynn's shot from close range was blocked by Jose Goncalves.
In a game that swung from end to end, Arfield's low drive from 18 yards on the hour mark was well saved by Balogh after the midfielder latched onto Flynn's knock down.
Falkirk manager Eddie May made the first change of the game when he replaced Flynn with Danijel Marceta in the 64th minute.
A minute later Falkirk striker Finnbogason mis-hit MacDonald's cut-back from 12 yards before Marceta launched the follow-up over the bar.
Superb trickery from Santana in the middle of the park set Hearts on an attack that ended when Stewart shot well over the crossbar from 25 yards.
Santana then had a shot easily saved by Olejnik in the 76th minute after Driver's cross was only partially cleared.
The frustration among the home hordes was palpable as the supporters sarcastically cheered and even booed Stewart's withdrawal from the game, when he was replaced by Ian Black.
Olejnik pulled off a great save to deny sub David Witteveen after the Austrian flicked Goncalves' long throw towards the far corner.
In injury time Falkirk defender Brian McLean produced a superb last-ditch tackle to block Witteveen's close-range shot as Hearts piled on the pressure, but Laszlo's men could not find a way through.