The Jambos had failed in their pleas to the SPL to postpone the match due to a virus which had laid low almost their entire team.
But despite this, John Robertson's side had only three changes from last Saturday's league match.
Inverness deserve all the credit for an excellent first season in the top league, which has now seen them defeat Hibs and Hearts and draw with Rangers in their last three matches.
Hearts made light of their problems with a good early attack and Darren Dods had to be lively to cut out Lee Miller's low cross from the right with Graham Weir lurking dangerously.
But it was Inverness that should had opened the scoring after 12 minutes when skipper Stuart Golabek's deep cross from the left found Roy McBain's head. Craig Gordon was beaten, but Andrew Webster popped up to clear it off the line.
In a dismal first half, the only other action of note was a 25-yard effort from Joe Hamill that whistled inches over the bar.
But Hearts should have broken the deadlock nine minutes into the second half when Dods' clearance broke to Dennis Wyness eight yards out, but he mistimed his jump and headed over the bar when he should have scored.
And that miss was to cost Hearts dear as the visitors went in front a minute later.
Richard Hart whipped in a free-kick from the right-hand side, and Dods was given the freedom of Tynecastle to glance a header beyond Gordon's despairing dive.
There was a scare for Inverness when Ross Tokely sliced a clearance against his own bar, but they wrapped things up in the 73rd minute and it was another defensive blunder from Hearts that did the damage.
Christophe Berra mis-hit his back-pass and Gordon lunged in to make an unnecessary challenge on Barry Wilson to gift Inverness a penalty.
Wilson picked himself up and stroked home the spot-kick to secure another great win for Craig Brewster's side.