Hearts tried to offload the Czech striker to Bristol City in the transfer window but the player refused to go.
How glad the Tynecastle side must be because they had fired blanks all afternoon until Pospisil made his timely entrance.
It was rough justice on Inverness, who had battled hard and made the clear chances in the match.
Indeed, Hearts had keeper Steve Banks to thank for a fine early stop when he threw himself to his right, to push away Graham Bayne's close-range volley.
Hearts might have had a penalty when Panagiotis Fyssas went down under Darren Dod's challenge, but it was Banks that kept them level with another fine stop.
Barry Wilson was off balance but fired in a 20-yarder that Banks managed to scramble around the post.
Hearts had another escape when Rory McAllister was given acres of space to head Richard Hastings' cross against the bar.
But Pospisil had the last laugh when he popped up at the back post to head Andy Driver's cross back across Michael Fraser and into the net to help Hearts secure all three points.