Carl Finnigan scored twice, and Mark Stewart and Scott Arfield also got on the scoresheet against Terry Butcher's side.
Inverness were reduced to 10 men in the 80th minute when goalkeeper Ryan Esson was sent off for bringing down Finnigan in the box, allowing Arfield to covert the spot-kick.
The victory for the Bairns breathes fresh life into their survival hopes as they took full advantage of a lacklustre Inverness display.
With the hosts arguably in greater need of the three points, John Hughes' side made a lively start as they dominated possession.
Patrick Cregg engineered an opening with a through-ball that found Finnigan but Lionel Djebi-Zadi recovered to get on the right side of the striker before clearing the danger.
Falkirk goalkeeper Dani Mallo was barely involved during the early exchanges, such was the home side's grip on the game.
Bairns winger Stewart was causing problems with his pace and should have done better with a shot which sailed well over the bar after finding space in the box.
Mallo breathed a sigh of relief after recovering to collect a loose ball, having initially dropped Ian Black's corner in the 23rd minute.
Inverness created their first real chance in the 26th minute when Adam Rooney rounded goalkeeper Mallo on the edge of the box but his shot from an acute angle went wide of the far post.
Inverness nevertheless looked dangerous on the attack and Richie Foran was afforded space in the box to connect with David Proctor's cross but his header bounced well wide of the target from 10 yards.
Falkirk then went close when Finnigan's cross from the left flank caused problems after Djebi-Zadi took a fresh air swipe at the ball at the near post.
The hosts then took the lead in the 37th minute through Finnigan.
Higdon teed up Jackie McNamara, who advanced from left-back, and the former Scotland international's dragged shot from inside the box fell into the path of Finnigan who slotted into the empty net.
With the Falkirk fans still celebrating, Inverness went close to levelling two minutes later after Mallo got down well to palm Dougie Imrie's 20-yard free-kick to safety.
In a frenetic start to the second period, Mallo flapped at Foran's 18-yard shot and Tam Scobbie came to his goalkeeper's rescue with Rooney about to pounce.
There was an urgency about Inverness' play following the loss of the first-half goal.
Imrie then stabbed the ball just wide at the near post after Rooney found his team-mate.
With Inverness looking dangerous, Falkirk dealt their opponents a blow with a second goal through Stewart in the 65th minute.
Mallo's long ball forward was flicked on by Higdon and Stewart used his pace to move clear before firing the ball beyond Esson.
Imrie had a chance to respond for Inverness but he lashed a volley well over from close range.
Inverness manager Butcher threw on another striker, bringing on Andy Barrowman for midfielder Russell Duncan, as the visitors looked to salvage something from the game.
Falkirk were dealt a blow with 10 minutes left when the experienced Steven Pressley was forced off injured after landing awkwardly.
But before substitute Gerard Aafjes could even replace the defender, the hosts were awarded a penalty and Inverness goalkeeper Esson was shown a straight red card.
Finnigan was brought down by the shot-stopper in the box after chesting Neil McCann's cross down.
Arfield slammed the penalty past substitute goalkeeper Michael Fraser to put the result beyond doubt.
Finnigan grabbed his second goal in stoppage-time to compound the misery for Inverness, meeting Cregg's pass at the near post to prod the ball past Fraser.