The turning point in the match came just after the half-hour mark when Lionel Djebi-Zadi was sent off for a professional foul on Carl Finnigan.
The Falkirk striker took the spot-kick himself and smashed the ball past Ryan Esson.
Falkirk have now made club history by reaching two cup semi-finals in one season for the first time.
Inverness made two changes to the side that defeated Rangers at Ibrox in midweek, with Eric Odhiambo starting in attack ahead of Adam Rooney, and Dougie Imrie returning as David Proctor missed out completely.
The Bairns made two enforced changes with Finnigan and Mark Stewart replacing Scott Arfield and Michael Higdon, both suspended.
Falkirk created the first opening when Arnau Riera's header fell kindly into the path of Steve Lovell but Esson raced out of his goal to thwart the former Aberdeen striker.
Ian Black, Caley Thistle's penalty hero from their midweek win at Ibrox, fired the home side's first attempt at goal harmlessly over the bar from a 30-yard freekick in the eighth minute.
Midfielder Black went much closer six minutes later when Dani Mallo was called upon to tip his left-footed shot out away for a corner, which the home side made little of.
The visitors were unlucky not to open the scoring when Lovell's header from Finnigan's cross fell the wrong side of the bar.
Then Richie Foran somehow failed to steer Imrie's inviting free-kick into the net from inside six yards at the other end.
After initially failing to gather an Imrie cross, Mallo then made a fantastic save to deal with Ross Tokely's looping header after 22 minutes.
Then came the incident which led to the goal.
Djebi-Zadi was caught the wrong side of Finnigan when Lovell flicked a cross from Burton O'Brien goalwards and the Frenchman recklessly bundled the Falkirk striker to the floor.
Referee Steve Conroy had no choice but to show the Caley full-back a red card and Finnigan smashed the penalty past Esson.
Manager Terry Butcher soon bolstered his Inverness defence to avoid falling further behind by bringing on Richard Hastings for striker Eric Odhiambo.
Falkirk could have doubled their advantage within 30 seconds of the second half starting but Patrick Cregg shot wildly over the bar from a good position.
Russell Duncan's shot went inches over the bar and Imrie was eventually stopped after a weaving run through the Falkirk defence as the Highlanders pressed for an equaliser.
The game looked set to spill over when Duncan fell to the ground after squaring up with Falkirk captain Darren Barr.
Both players were booked, as was Mallo for sprinting half the length of the pitch to join in.
With Inverness seeing their Scottish Cup dreams hanging by a thread, Butcher went for broke by bringing on Rooney for Filipe Morais after 75 minutes.
The home side threw everything at the away defence in the closing stages and Brian Kerr, who had come on for Black after 57 minutes, came close with a powerful strike that Barr did brilliantly to head clear to safety.
Even full-back Tokely was pushed up front as an additional targetman for the final 10 minutes as the Inverness onslaught continued.
Great stops from Mallo denied Imrie and then Foran, preventing the Highlanders from forcing a replay.