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.
5. Hearts' Jorge Grant is kept at bay by Zurich's Blerim Dzemaili
Hearts' Jorge Grant is kept at bay by Zurich's Blerim Dzemaili Photo: Ennio Leanza
6. Jorge Grant plays a pass under pressure from Zurich's Blerim Dzemaili
Jorge Grant plays a pass under pressure from Zurich's Blerim Dzemaili Photo: Ennio Leanza
7. Zurich's Donis Avdijaj is put under pressure by Peter Haring
Zurich's Donis Avdijaj is put under pressure by Peter Haring Photo: Ennio Leanza
8. Peter Haring is wrong-footed by Zurich's Donis Avdijaj
Peter Haring is wrong-footed by Zurich's Donis Avdijaj Photo: Ennio Leanza