Having been on the end of a 4-0 defeat against Rangers at Tynecastle only three days, the Jambos were keen to keep their hopes of a top-six finish alive with a victory over fellow hopefuls Caley Thistle.
Inverness, on the other hand, were looking to end a run of six games without a win, but fell behind after 22 minutes.
Early on Markus Paatelainen was relieved of possession next to his own corner flag and Lee Wallace drove into the box, before setting up Christian Nade, who watched his low drive superbly saved by Fraser.
From the resultant corner, however, Andrew Driver found the unmarked Christos Karipidis and he made no mistake from six yards out.
Ten minutes later and Hearts doubled their lead, Nade used his pace and power to leave both Grant Munro and Russell Duncan in his wake as he crossed for strike partner Elliot to fire a low shot past Fraser.
Most of Caley Thistle's attacks had broken down in the final third, although Don Cowie was unlucky to see his 20-yard free-kick on 36 minutes deflected wide.
Marius Niculae also came close to moving his goal tally into double figures for the season two minutes from the interval, but he just failed to connect with Ross Tokely's low cross.
Inverness manager introduced Dennis Wyness for the ineffective Paatelainen at the start of the second half, but it made little difference as Hearts moved into a three-goal lead on 47 minutes.
Ian Black's short back-pass put Fraser in all sorts of trouble and as Tokely came in to try and clear up the mess he only succeeded in turning it into the path of the grateful Elliott who showed plenty of composure to loft the ball into the open goal.
Caley Thistle were offering little in attack and it took them until 75 minutes to muster their first shot on target, but Niculae's effort went straight into the arms of Hearts goalkeeper Steven Banks.
The visitors, however, were rarely troubled throughout the 90 minutes as they cruised to a fine win.