The faintest of touches from the striker on a David Templeton cross nine minutes from time was enough to guide the ball in off the post.
Caley Thistle forward Gregory Tade had broken the deadlock five minutes after the interval with an angled 12-yard effort following a mistake from goalkeeper Jamie MacDonald.
Tenth-placed Inverness could have closed the deficit on fourth-spotted Hearts to one point if they had held on, but the Edinburgh club might have stolen the victory late on if Andy Webster had connected with a Jamie Hamill corner.
Midfielder David Davis was inserted straight into the Caley Thistle team after joining the club on loan from Wolves on transfer deadline day.
The 20-year-old had an impressive debut alongside Andrew Shinnie - the only other change for Terry Butcher's men from the side which won 2-1 against Kilmarnock last time out, with Aaron Doran and Chris Hogg not involved.
Paulo Sergio also made two changes to the starting line-up which beat Hibernian 2-0 in their last outing, goalkeeper Marian Kello replaced by MacDonald and Arvydas Novikovas in for Andrew Driver.
Inverness started brightly with Graeme Shinnie and Davis combining well to set up captain Richie Foran, who was thwarted at close range by a sliding Daniel Grainger tackle.
It was the only notable opportunity in the opening stages as the sides cancelled each other out with neither goalkeeper coming under any sort of pressure.
Webster would have opened the scoring for Hearts on 25 minutes had it not been for Graeme Shinnie stopping his shot on the line following a Hamill corner.
That sparked the visitors briefly into action with Elliott testing goalkeeper Ryan Esson after coming onto a Ryan Stevenson lay-off. The stopper easily blocked the shot from 12 yards.
Inverness fought back with the Shinnie brothers working together to set up Foran, whose header flew inches wide.
The opening goal arrived five minutes after the break. Caley Thistle midfielder Nick Ross cashed in when MacDonald slipped and placed the ball into the path of French forward Tade, who hammered home his second goal of the season.
The attacker might have doubled the lead five minutes later when he latched onto an Andrew Shinnie through-ball, but MacDonald saved.
Hearts went close to levelling on the counter-attack when substitute Templeton hit the crossbar, but Inverness hit back with Davis breaking down the wing before crossing for Andrew Shinnie who sent the ball just wide.
The onslaught from the hosts continued, with MacDonald equal to Tade again after he was played in by Foran.
Esson saved well from Grainger following a Hamill corner as Hearts searched for an equaliser.
MacDonald made another important save to keep the visitors in the game when he blocked Foran's low shot.
The Irishman's angled effort came after good work from fellow forward Tade, whose pinpoint deep cross found him with time and space but the keeper stood firm with 15 minutes to go.
Hearts got back on level terms in the 81st minute when Elliott got a touch on a Templeton cross to send it past Esson via a post.
Webster might have snatched a last-gasp winner for Hearts but he failed to meet Hamill's late corner which flashed across the face of the goal.
Inverness also had time to come close at the death but substitute Aidan Chippendale's shot went just wide in stoppage time.