Hapoel were under the cosh for much of the game, but had looked set to claim a creditable goalless draw thanks to an inspired performance by keeper Daniel Amos.
The Hapoel stopper produced a breathtaking save to deny Ricardo Nunes from the penalty spot with just nine minutes remaining as the visitors clung on for dear life.
But just 60 seconds later, Zilina finally made the breakthrough when Piaček converted Miroslav Barčík's corner.
Amos had already produced a number of outstanding saves with Momodou Ceesay, Nunes and Piaček all denied before Zilina’s late strike.
Hapoel did create chances of their own with Shimon Abhazira shooting inches wide, while Sali Hassarmi struck the bar when he looked set to claim a crucial away goal.
The Israeli champions will now hope to overturn the deficit when the teams meet again next week.