Andrew Shinnie had made a lively start for Inverness in the opening period of the game before setting up the opening goal for Tudur Jones after 19 minutes.
Shinnie did well before passing the ball to the 28-year-old, who fired the ball into the top corner from the edge of the penalty area.
The home side continued to press forward and they could have doubled their lead, but Richie Foran saw his shot go over the crossbar.
That appeared to be the catalyst for Hearts to become more involved in the match as Ryan Stevenson quickly saw a shot saved.
Hearts continued to press the Inverness goal but they were unable to equalise before the half-time whistle.