The home side hit the woodwork four times in the first half and produced a barrage of shots on target while many more went high or wide as they failed to capitalise on their relentless waves of pressure against a team they had beaten with nine men a fortnight earlier to reach the League Cup final.
However, nothing could disguise Gary Locke's pleasure at taking a point as he brushed aside speculation about his future which has suggested Hearts' prospective new owner, Ann Budge, would bring back Craig Levein as manager.
Locke said: "There is still a long way to go Bryan Jackson, the administrator, is saying as much. But it is great we are going to have a new owner who is going to take the club forwards in the right direction.
"That's fantastic for everybody. Everything over the last few days [about his position] has been pure speculation. The big thing for me is this is the first bit of good news we've had in a long, long time.
"We were in a position in the summer where it was probably as bleak as it gets. The club was on its knees and there was a serious chance of us not being here. Now, hopefully, we have a right good chance of a really big future. That's the only thing that concerns me."
The hosts were boosted pre-match by the announcement midfielder James Vincent signed a contract extension through to summer 2016.
Hearts might have been fired up by thoughts of revenge after that League Cup semi-final defeat but instead they were left chasing shadows as Inverness dominated.
Marley Watkins was at his menacing best on the right flank, frequently cutting inside and around the back to test the Hearts defence, and came within a whisker of scoring with a low strike after 10 minutes.
Full-back David Raven's attempted cross swirled on to the bar soon after and Danny Wilson had to block desperately to stop Billy McKay converting the rebound.
The chances kept coming and it was a small miracle that the scoreline remained level. Gary Warren arced a header just over from Graeme Shinnie's corner while another Watkins run and cross was followed by Jamie MacDonald leaping to clutch a powerful Greg Tansey header.
The goalkeeper shut down a point-blank McKay shot inside the box before the half hour and then tipped a Watkins header on to the bar. McKay's overhead kick from the rebound bobbled off the top of the crossbar and then Greg Tansey thumped the post.
There was more urgency and intent from Hearts after the break with Jamie Hamill sending a 25-yard attempt just wide.
Another Tansey strike from distance was brilliantly touched for a corner by MacDonald.
Hamill might even have won it for Hearts as he rocketed one shot inches over and another attempt just past a post late on.
"I felt in terms of the style in which we want to play football, it was all there today, a bit of everything," said Inverness manager John Hughes.
"The shape, passing and energy levels were very good. We hit the woodwork four times. On another day, we could have won it handsomely, but it wasn't to be."