Driver pounced on poor defending before darting clear of the Bairns defence and rounding the goalkeeper in the 69th minute at Tynecastle.
Falkirk marksman Steve Lovell gave the visitors the lead in the 16th minute after being played through by Michael Higdon but Hearts equalised just three minutes later with a superb 20-yard free-kick from Aguiar.
Hearts were keen to stamp their authority on the game early on and created several credible chances, the first in the fifth minute when Christian Nade teed up Bruno Aguiar on the edge of the box but the midfielder dragged his shot wide.
Moments later, Lee Wallace caused mayhem in the Bairns defence when he rode several tackles on the byline before setting up Nade but the Frenchman's close-range effort was blocked by Kevin McBride and Eggert Jonsson's follow-up was also deflected clear.
Hearts had their visitors pegged back and Aguiar volleyed just inches over from 16 yards after the ball fell to him from Robbie Neilson's long throw.
However, Falkirk shocked their hosts by taking the lead in a rare attack in the 16th minute.
Michael Higdon slipped a pass through to Lovell and the striker ran clear of the defence before toe-poking the ball past Janos Balogh.
Their lead lasted a mere three minutes, however, as Aguiar planted a 20-yard free-kick in the top right corner of the net after Darren Barr's foul on Nade.
Despite surrendering their lead Falkirk began to settle and started stringing together passes.
Chris Mitchell tried his best to emulate Aguiar from a dead-ball situation but his free-kick from 25 yards just bounced past Balogh's post.
At other end, Driver used his pace to good effect by bursting into the box but his teasing pass across goal was just behind Jonsson, who had made a late run from midfield.
Aguiar again caused problems from a set-piece 10 minutes before the interval after his delivery from the near side was met by Marius Zaliukas but the LIthuanian failed to keep his header down.
Hearts defender Lee Wallace became the first player in the match to be cautioned by referee Eddie Smith after a rash challenge on Kevin McBride minutes after the restart.
The home fans thought their team should have been awarded a penalty moments later for a handball by Lee Bullen from Driver's cross but Smith waved play on.
Driver was involved in Hearts' next chance after creating room for himself on the flank to find Christos Karipidis in the box.
The Greek took a touch before hitting a left-footed shot that swung just wide of Scott Flinder's post.
Driver then gave Hearts the lead in the 59th minute with a superb solo goal. The pacy winger picked up Bullen's slack pass and sped away from Barr before cooly rounding Flinders to find the net.
The game opened up as Falkirk left gaps at the back by pushing men forward in search of an equaliser and Mitchell, who has an earlier gone close with a free-kick, shot from 40 yards out on the touchline and Balogh was forced to tip over for a corner.
From the resulting set-piece Aguiar headed Bullen's header off the line from Mitchell's delivery. And Hearts survived an injury-time scare when Higdon headed wide from Mitchell's cross.