Inverness Caledonian Thistle 0 Falkirk 1
Inverness were keen to build on their surprise win at Rangers in midweek, but it was the Bairns who had the first chance inside two minutes when Steve Lovell latched on to a hopeful throughball only for goalkeeper Ryan Esson to make the block.
The home side responded after 14 minutes when Ian Black tested Falkirk number one Dani Mallo with a 20-yard shot that the goalkeeper turned away, while at the other end Lovell's header from Carl Finnigan's cross landed on the roof of the net five minutes later.
The opportunities were now coming thick and fast as Caley Thistle almost took the lead a minute later, but neither Richie Foran nor Grant Munro could get themselves on the end of Ross Tokely's header across goal.
Tokely was then unlucky not to score himself on 22 minutes, taking advantage of a mix-up between Steven Pressley and goalkeeper Mallo from Dougie Imrie's cross to send an ambitious looping header towards goal from around 25 yards that the Bairns keeper only just got back to claw away.
It was all Caley Thistle now, but a moment of stupidity from Lionel Djebi-Zadi handed Falkirk the crucial goal on the half-hour mark.
There looked to be little danger from a cross into the box, yet the Frenchman was clearly manhandling Finnigan as the ball bounced through to Esson.
To make matters worse, Djebi-Zadi was red carded by referee Steve Conroy and his misery was only compounded when Finnigan sent Esson the wrong way from the spot.
Falkirk then spurned a decent chance to put the game beyond Inverness barely a minute into the second half, but Patrick Cregg fired high over the crossbar after being set up by Mark Stewart.
Despite being down to ten men, Caley Thistle still posed an attacking threat and Russell Duncan was close with a dipping shot from the edge of the box after 50 minutes following good work by Foran and Imrie.
Duncan was then guilty of passing up another opportunity on 65 minutes as Mallo struggled to clear a cross, but he skewed his effort well wide from a good position.
In stoppage time, Foran latched on to Tokely's flick-on only for Mallo to race from his line and make a wonderful block to deny the Irishman as Falkirk progressed into the semi-finals.