The visitors created all the clear-cut chances in a poor match at Westfield but could not find a way past Bairns keeper Dani Mallo on an afternoon of frustration.
Andrew Driver was at the centre of almost all of Hearts' best moments but was guilty of passing up a magnificent first-half chance on the break when he shot wide with only Mallo to beat.
Mallo also touched over a David Obua header and Marius Zaliukas somehow contrived to miss an open goal from only three yards as Csaba Laszlo's outfit could not find a route to the net.
Hearts, who lost goalkeeper Janos Balogh through injury from the side which drew 1-1 with Celtic the previous week, were first to threaten in a lively opening between two teams eager for all three points for very different reasons.
Obua made a telling impact as he leapt highest to flick the ball on to Driver but the winger's glancing header was deflected wide for a fruitless corner by Jackie McNamara.
Bottom side Falkirk enjoyed plenty of possession in the early exchanges, with manager John Hughes being handed a vote of confidence from chairman Campbell Christie ahead of the game following calls from fans for him to be sacked.
But it was the visitors who should have taken the lead on the break twice in the opening 12 minutes.
Calum Elliot did brilliantly deep in his own territory to spark a seventh-minute move involving Bruno Aguiar and Lee Wallace before Driver neglected the opportunity to shoot and played a poor lay-off to Obua that was intercepted by the scrambling Bairns defence.
Six minutes later, Neil McCann was robbed by Aguiar on the halfway line but Driver whipped a left-foot drive wide with only Mallo to beat.
In the 20th minute, the home goal survived another scare.
Ruben Palazuelos headed on Aguiar's free-kick, and when Michael Higdon inadvertently set up Zaliukas three yards out, the Hearts defender somehow contrived to miss an open goal.
Hearts almost produced the breakthrough on another lightning-quick break in the 28th minute.
Aguiar found Driver wide on the right and the youngster's cross was perfect for Obua but the Ugandan's downward header was scooped over the bar by Mallo.
Referee Eddie Smith was in danger of losing control of proceedings as he was forced to dish out warnings to Falkirk management team John Hughes and Brian Rice and yellow cards to Arnau Riera and Aguiar following bookings for Palazuelos and Higdon.
Falkirk began the second period in more determined mood and in the 49th minute at least forced a save out of stand-in Hearts keeper Marian Kello but Higdon's header from Burton O'Brien's cross was too tame and straight to trouble the Slovakian.
In the 66th minute, Hearts launched another rapier-like counter-attack which should have resulted in the opener for the visitors.
Lee Wallace careered forward from left-back and slipped the ball to Driver but his right-foot shot after cutting inside was straight at Mallo.
Patrick Cregg shot over for Falkirk minutes later as Hearts showed their intent by removing midfielder Michael Stewart in favour of striker Christian Nade.
Both teams pressed for what looked likely to be the winner in the closing stages but it was the visitors who came closest.
First, Nade had an attempt to turn in Driver's cross at the near post blocked by Steven Pressley before Obua fired over from Driver's cutback on the break.
With five minutes remaining, Aguiar had a netbound effort blocked by Riera and then saw his free-kick headed unconvincingly over by Nade.