The home side did not manage a shot on target over the 90 minutes but will take heart from the fact they didn't concede a goal.
Hearts fashioned an early chance when Andrew Driver managed to get his head to the ball in a crowded area but the ball broke off a defender and finished wide.
Falkirk's response was swift and a deep cross from Neil McCann found Michael Higdon arriving late at the back post but he could not connect properly and the chance was lost.
Driver wasted a good chance to put Hearts ahead after McCann was robbed by Bruno Aguiar in midfield. Aguiar set up his team-mate but Driver had too much time to think about it and steered his left-foot shot a foot wide.
There was another let-off for Falkirk when they failed to deal with an Aguiar free-kick and the ball broke to Marius Zaliukas just a yard from goal but, at full stretch, he could not apply the finish.
Hearts looked the more likely going forward and Dani Mallo had to react to flip a David Obua header over the top after Driver had provided the cross.
Aguiar then sent a 25-yard shot well wide after Calum Elliot had knocked down a throughball from Zaliukas.
Driver had another opportunity after Obua headed the ball down but Arnau Riera managed to get a touch at the vital time.
Falkirk did not have an attempt on goal in the first half but there was an early chance for Higdon just after the break after good work from Burton O'Brien but he could get no power on his header.
The home side pressed more but it was Hearts who should have taken the lead midway through the second half after a great break by Lee Wallace.
He laid the ball to Driver but the winger's shot was directed straight at Mallo and Falkirk were let off the hook again.
Falkirk were similarly wasteful moments later when Patrick Cregg had a shooting chance when the ball was rolled to him at the back post but he was leaning back and sent his effort yards too high.
Some of the finishing was woeful. Elliot lashed a 20-yard effort high and wide after Driver had set him up.