Keen to improve on a home record that showed only one win from seven SPL games this season, Inverness enjoyed a bright opening with Andrew Barrowman only inches away from connecting with a Barry Wilson cross.
Hearts were showing little threat in the final third, but they stunned the home crowd into silence on 22 minutes.
Bruno Aguiar was given far too much time and space 30 yards from goal and he used it to good effect, sending in a low drive that angled away from goalkeeper Ryan Esson's reach and nestled in the corner.
It all went a bit flat for Inverness following the loss of the goal, although they should have gone in at the interval on level terms.
Substitute Roy McBain, on early for the injured Ian Black, sent Andy Barrowman racing clear and, having brushed aside the challenge of goalkeeper Janos Balogh, he looked favourite to score.
Unfortunately for the striker, neither he nor fellow forward Adam Rooney could force it over the line as Robbie Neilson made a timely interception.
Hearts started the second half the better and twice Esson was forced into smart saves, firstly from Christian Nade and then Andrew Driver.
Slowly, Caley Thistle began to edge back into proceedings but, despite some sustained pressure in and around the visitors' box, they still had yet to seriously test Balogh.
Dougie Imrie attempted to change all that with a 20-yard shot that skidded a yard wide of the Hearts goal.
A late effort by Iain Vigurs finally tested Balogh late on, but it was not enough to breach a Hearts rearguard that made it three clean sheets in succession.