The share of the spoils means Inverness remain ninth, one place above their midweek visitors, who took at 19th-minute lead through Michael Higdon.
Caley Thistle, seeking only their second home win of the season, almost got off to the perfect start when a sliced clearance fell to Dougie Imrie, who drove a shot wide from 20 yards in the opening attack.
In the early stages, Falkirk were looking to use width, with Lee Bullen advancing down the right and Neil McCann threatening on the opposite flank.
Russell Duncan tried to test visiting keeper Scott Flinders from distance in the ninth minute, but his effort flashed wide.
The opening exchanges were peppered with needless challenges as both sides tried to get a grip on the game.
With the table so tight, these sides knew that three points could push them right up the order.
The Bairns stole in the lead in the 19th minute when McCann's deep delivery was not dealt with by the defence and Higdon finished after Stevie Lovell nodded the ball into his path.
Falkirk could have added another moments later but Esson gathered Higdon's low drive.
Inverness responded well, though, and Don Cowie fired just past the right post after fine work on the right by Iain Vigurs and Imrie.
Flinders gave John Hughes' men a few scary moments as he was less than convincing when dealing with crosses on more than one occasion.
Higdon came close to notching his second after 35 minutes when he beat the offside trap but drove wide from 12 yards.
On the stroke of half-time, McCann had to go off after sustaining a knock and was replaced by Graham Barrett.
To freshen up his attack at the start of the second half, Inverness boss Craig Brewster gave the nod to subs Andy Barrowman and Rooney who replaced Vigurs and Garry Wood.
It was Falkirk who carved out the first chance of the second period, though, and Barrett should have done better than head well wide from Lovell's cross from the right.
Flinders was called into action on 53 minutes and the big keeper did well to deny a close-range pop from Imrie.
Caley Thistle appealed for a penalty on 62 minutes when Ian Black went down in the box under a challenge from Jackie McNamara, but play was waved on.
In a final throw of the dice for the hosts, Barry Wilson was introduced with Cowie coming off.
However, the Bairns were pushing for their killer second and Higdon went close when he steered a header wide from Dean Holden's delivery.
The defining moment for the Inverness troops came in dramatic fashion.
With nine minutes to go, Caley Thistle grabbed the leveller when Rooney reacted quickest in a penalty box to net after team-mate Black crashed an effort off the bar.
Despite both sides going all out for the winner at the death, each had to settle for a point apiece at the full-time whistle.