Robbie Neilson’s side have given themselves a huge opportunity to reach the lucrative group stage, despite surrendering an early lead in St Gallen.
Lawrence Shankland's penalty put Hearts ahead in the play-off first leg in St Gallen, but Zurich scored twice in two minutes before the break through Adrian Guerrero and Blerim Dzemaili.
And although Hearts mustered a stronger finish, goalkeeper Craig Gordon produced a vital save in the last minute to keep the deficit at one goal.
And although Hearts mustered a stronger finish, goalkeeper Craig Gordon produced a vital save in the last minute to keep the deficit at one goal.