Substitute James Vincent scored on 86 minutes shortly after Peter Grant levelled for Falkirk to cancel out the opening goal from Marley Watkins.
Caley Thistle defender Carl Tremarco was dismissed on 75 minutes, but the Highlanders showed their mettle to claim the prize only 21 years after the club was formed.
It was another milestone for John Hughes’ side who had already qualified for Europe next season for the first time due to their club-record third-place finish.
Thistle almost opened the scoring on nine minutes when Aaron Doran got a flick on a cross from Graeme Shinnie and the ball just eluded Watkins as he arrived unmarked in the centre of the penalty area.
But Inverness could have been punished 11 minutes later when they lost possession of the ball just outside the penalty area trying to pass the ball under pressure and Will Vaulks drilled a low shot from 15 yards which had Ryan Esson scrambling down at his post to save.
Vaulks had some of the Falkirk fans on their feet cheering when he crashed another shot from distance into the net, but this time the ball was the wrong side of the post and rustling only the side-netting.
Watkins then timed his run to perfection on to a lofted pass but Grant got across to make a sliding block, and from the resulting corner which Falkirk struggled to clear Edward Ofere nodded the ball back into the box only for Doran to fluff his attempted volley.
But the opening goal arrived seven minutes before the interval as Shinnie drifted in from the right and slipped the ball to Ofere to bring Doran into play. He threaded a pass ahead of Watkins and the Caley midfielder retained his composure to round the advancing Jamie MacDonald and slot home.
Grant came to the rescue again in the last minute of the half with another sliding challenge to take the ball off the toes of Doran as he raced through on goal.
Seven minutes after the break Rory Loy laid the ball off to Vaulks and darted between two inverness defenders for the return, but the Falkirk defender let fly from 25 yards, pulling his shot just wide of the target.
And the Championship side had another half-chance on the hour when Keiran Duffie raced down the right touchline scooped the ball towards the penalty spot where Blair Alston arrived to hook his shot over the bar.
Vaulks again came close when a deep corner dropped at his feet and he side-stepped a challenge before lifting his shot just over the bar.
With 15 minutes remaining Alston robbed last-man Tremarco and was brought down by the Caley Thistle defender, who was red carded by referee Willie Collum.
Falkirk had been pushing their opponents back for most of the second half and now had the extra man.
Five minutes later Alston delivered a free-kick from the left side of the Caley Thistle box where Grant arrived at the back post to power his header under the bar to level the match.
Falkirk had all the momentum but they were caught out by the pace of Watkins who ran from just inside his own half and tried to place his shot into the left corner, but while MacDonald got down to parry the ‘keeper pushed the ball straight in front of Vincent to tuck it away.