Caley Thistle, unbeaten away from home since November 2009, went in front on their final road trip of the year through Grant Munro's 17th-minute header.
Kevin Kyle equalised from the penalty spot after 25 minutes, but the hosts missed the chance to move level with Celtic in the table as Caley Thistle's determined rearguard frustrated Hearts, extending the visitors' run to six games without defeat.
With Celtic's match at Hamilton postponed, manager Jim Jefferies knew victory would take his Hearts side level in second place in the table, although the Bhoys now have two games in hand.
Ian Black and Eggert Jonsson recovered from illness to allow Jefferies to revert to the XI which began last week's 5-0 defeat of Aberdeen.
Hearts defeated Motherwell 2-1 on Tuesday night, taking their run to 15 goals scored and one conceded in securing six wins from six and eight out of nine.
The hosts had the better of the opening period and Stuart Duff received a yellow card for a cynical challenge on David Templeton, who was racing down the left with team-mates queuing up in the centre.
Caley Thistle half-cleared the resulting free-kick with Black, who signed for Hearts from Inverness in June 2009, firing over with a left-foot shot from the edge of the area.
The visitors went in front after Russell Duncan's inswinging free-kick from the left found Munro at the far post.
The defender's header towards goal created havoc in the Hearts defence - a rearguard temporarily without Ismael Bouzid - and evaded everyone to bounce beyond the line.
Bouzid was holding his back and was replaced two minutes later, with Ryan Stevenson coming into midfield and Adrian Mrowiec switching to centre back.
Templeton was booked for a dive after going down under the challenge of Duff in the penalty area but then Hearts were awarded a spot-kick when Stephen Elliott was brought down by goalkeeper Ryan Esson as he attempted to latch on to Kyle's knock down.
Esson was booked by referee Brian Winter for the challenge before Kyle sent him the wrong way with a successful spot-kick.
Hearts went close twice before the interval, with Templeton firing a stunning 25-yard strike against the post before Esson saved Stevenson's left-foot shot from the edge of the area.
The hosts continued to dominate possession and territory after the break, but Caley Thistle looked dangerous on the break.
Templeton shot wide across goal after 50 minutes after streaking down the left before Adam Rooney had an effort ruled out for offside after 52 minutes.
Kyle then brought a long ball down and turned towards goal but could not get a shot away under Ross Tokely's challenge and Inverness cleared.
Jonsson's cross was then directed towards goal by Kyle, but Esson gathered at the second attempt.
Templeton tried his luck from long range again, the ball swerving narrowly wide.
Skacel, who had been a peripheral figure, then shot a yard wide across goal after ghosting into the penalty area unchallenged.
Skacel was played in down the left by Stevenson nine minutes from time but the Czech midfielder's ball across goal flashed in front of Kyle and Elliott and Caley Thistle survived again.
Duncan shot over for the visitors before Elliott was replaced by Calum Elliot with seven minutes remaining.
Kyle blasted over before Inverness scrambled clear a series of Hearts corners.
Duncan broke in stoppage time and forced a save out of Marion Kello, but the match ended level.