Steve Lovell's hotly-disputed 59th-minute winner saw the Bairns emerge victorious in the fifth-round tie at Tynecastle after both teams finished with 10 men.
Hearts had look the side most likely to break the deadlock before Lovell struck, with the visitors thankful to a superb display in nets from Dani Mallo.
Further controversy raged soon after Lovell's goal when referee Dougie MacDonald awarded Hearts a penalty only to change his mind after consulting with his assistant.
Marius Zaliukas and Scott Arfield both saw red late on - nine cautions and two reds were dished out in all - as John Hughes' men survived to seal a fine win.
It was Hearts who carved out the first opportunity. Andrew Driver's neat backheel allowed left-back Lee Wallace to burst into the penalty area and his angled drive was palmed away by Spaniard Mallo.
Falkirk were being quickly pinned back, even if Neil McCann was looking a threat.
His fellow former Hearts star Steven Pressley was the victim of some surprising booing from home fans, despite leading the club to Scottish Cup glory in 2006.
Hearts' dominance almost yielded the breakthrough after 23 minutes. The lively Driver was the catalyst, cutting back for Gary Glen, who side-footed just wide.
Michael Higdon had struck a limp shot at the recalled Janos Balogh early on, but it was 27 minutes before they again registered an effort when Dean Holden dragged wide.
The game became a scrappy affair until, eight minutes from the interval, Falkirk came close to breaking the stalemate.
Holden's lofted free-kick saw Michael Higdon beat the onrushing Balogh to the ball, but the forward's header drifted just past. Glen replied with a low drive comfortably held by Mallo.
The Bairns' new custodian then produced even greater heroics on 42 minutes.
Bruno Aguiar's superb pass played in new captain Robbie Neilson on the right and his cutback from the byline saw Glen strike fiercely goalward, only for Mallo to brilliantly tip over.
Bairns captain Darren Barr picked up a head injury moments into the second half and the visitors were temporarily down to 10 men as he went down the tunnel to be patched up with a bandage.
Hughes made a change as early as the 56th minute in the search for a goal, withdrawing Arnau Riera for Carl Finnigan. Higdon then flashed over from 18 yards two minutes later.
The game came to life after 59 minutes when Falkirk took the lead in controversial circumstances.
Jackie McNamara's lofted ball into the box saw Lovell beat Balogh with his header and the ball hit the underside of the bar before dropping over the line, with Finnigan making sure.
Despite fierce protests from Neilson in particular, claiming Lovell was offside, referee McDonald let the goal stand.
Higdon then immediately had the ball in the net again, but was this time ruled offside when it appeared a close call.
In a frantic period of play, Hearts broke straight up the park and Driver was brought down by Holden near the byline, with McDonald awarding a penalty.
However, after consulting with his assistant following Falkirk protests, the whistler changed his mind and instead awarded a corner to the fury of the majority in Tynecastle.
Falkirk scrambled clear the set-piece before - in a now breathless contest - going close themselves through a Barr header.
Laszlo took off Jonsson and Glen for Laryea Kingston and Calum Elliot in search of the leveller, with Aguiar soon driving just wide.
Mallo then superbly tipped over from under his bar after McCann had inadvertently prodded back to his keeper.
From the corner, Pressley superbly blocked out Christos
Karipidis at close range.
Amid the mayhem, Barr was again forced to go off for treatment with blood from his head wound.
Hearts continued to press and Mallo was relieved to see Aguiar's free-kick curl wide. But the home cause was hindered eight minutes from time after a moment of madness from Zaliukas.
The Lithuanian hauled down Finnigan for a free-kick on the edge of the area, but then punched the ex-Newcastle striker on the ground. McDonald brandished a straight red card for the defender.
Kingston almost rescued his team when he hooked wide at the back post from a Karipidis header back across goal.
At the start of five minutes of stoppage-time, further drama followed when Arfield was shown a second yellow card for a challenge on David Obua to leave both sides with 10 men.
Pressley was then lucky only to be cautioned when he hauled down Michael Stewart as he surged in on goal.