The visitors dominated from the first whistle and there was no surprise when the Irishman headed the Gorgie side into the lead in the 22nd minute from a Lee Wallace cross.
The first 45 minutes showed why Accies had failed to win a home league game this season and that record was kept intact a minute after the break when Elliott deflected in a Marius Zaliukas header from a Rudi Skacel corner.
Billy Reid's men had looked forlorn in the first half when they were completely outplayed and, although they improved after the interval, in the end it proved to be one of Hearts' easiest wins of the season and provides further evidence for those who believe the Lanarkshire side are on the slide.
Pre-match interest surrounded Andy Webster, who was making his second debut for the Tynecastle side.
The big defender, who returned to his former club after being released by Rangers on transfer deadline day, replaced Ismael Bouzid while Wallace came in for Adrian Mrowiec.
While Webster could have spent the first half back at Ibrox watching Rangers play Motherwell, such was the lack of threat from the Lanarkshire side, Wallace was key to the Jambos' superiority.
The Scotland left-back helped the Gorgie men dominate possession as Accies shaped up in a defensive formation which hinted at a long afternoon for the suffering Hamilton fans.
Mickael Antoine-Curier was Accies' lone striker and he had virtually no support in the opening stages as Hearts kept the ball albeit without threatening New Douglas Park goalkeeper Tomas Cerny.
In the 16th minute Jim McAlister dispossessed Wallace as he raced down the left but the home side's promising break, their first of the match, ended with the Accies midfielder overplaying his cross to the back post.
Moments later, at the other end, Tynecastle midfielder Eggert Jonsson headed over from a Skacel corner.
There was almost an inevitability about Elliott's opener after 22 minutes.
Jonsson fed Wallace down the left and when another great cross from the Gorgie defender came into the box the Irishman dipped his headed to flick the ball past Cerny from 12 yards.
It was just reward for the Edinburgh team's superiority and a sorry looking Accies side seemed incapable of responding.
Just after the half-hour mark, as Hearts swarmed around the Hamilton box again, defender Craig Thomson drilled a low shot just past the far post.
Then, Jonsson was at the back to post to meet a lofted Thomson corner from the left but volleyed high over.
Hearts did grab their second in the opening moments of the second half, Elliott deflecting in a Zaliukas header from Skacel's corner.
In the 73rd minute McAlister had another left-footed effort from the edge of the box but Marian Kello confidently smothered.
Hearts clearly thought the points were in the bag as the game entered its closing stages - to the annoyance of their bench, who urged an increase in energy.
Accies huffed and puffed but the goal did not arrive.