Hearts had an early scare when Steve Lovell sent a dangerous ball across the face of goal but Ruben Palazuelos was able to hook the ball to safety.
Christian Nade set up a shooting chance for Bruno Aguiar but he sent his 20-yard effort well wide.
Lee Wallace then tore the Falkirk defence to shreds with a forceful run into the area but Nade and then Eggert Jonsson saw shots blocked as they sought to finish the move off.
It was a positive start from Hearts and Aguiar was just a foot too high with an 18-yard volley after Nade had headed on a throw-in from Robbie Neilson.
Falkirk were hemmed in their own half for the opening stages but broke with devastating effect to take the lead after 16 minutes.
Michael Higdon slipped the ball through to Lovell and he toe-poked the ball past the out-rushing Janos Balogh to find the corner of the net.
But the lead lasted just two minutes. Darren Barr was penalised for climbing all over Nade to give Hearts a free-kick 25 yards out.
Aguiar stepped up to curl a right-foot shot over the wall and, although Scott Flinders managed to get his hands to it, he could not prevent the ball finding the top corner of the net.
The game was being played at a furious pace and Higdon again showed his threat when he sent a decent shot over the top from just outside the box.
Chris Mitchell then had Balogh scrambling across his goal with a low free-kick but the ball slipped just wide of the goalkeeper's right-hand post.
Hearts started the second half well and almost took the lead after 55 minutes. Driver cut the ball back for Hristos Karipidis and his 15-yard shot was just inches wide of the far post.
But the goal was only delayed four minutes. Arnau Riera misplaced a pass in midfield and Driver pounced to waltz past the Falkirk defence and take the ball around Flinders before sliding it into the net.
Hearts kept the pressure on and Nade sent a header straight at Flinders after Falkirk struggled to clear another Neilson throw-in. Burton O'Brien clipped a 20-yard shot over the top at the other end as Falkirk tried to get something from the game.
Balogh had to be on his toes to touch a Mitchell lob over his crossbar and then Aguiar headed a Lee Bullen header off the line from the resultant corner.
Hearts lived dangerously in the final minutes with Higdon heading wide from close in but they held on for an important victory.