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.
1. Zurich's Nikola Boranijasevic and Alex Cochrane collide at Kybunpark
Zurich's Nikola Boranijasevic and Alex Cochrane collide at Kybunpark Photo: Ennio Leanza
2. Toby Sibbick and Cammy Devlin try to chase down Zurich's Wilfried Gnonto
Toby Sibbick and Cammy Devlin try to chase down Zurich's Wilfried Gnonto Photo: Ennio Leanza
3. Zurich's Blerim Dzemaili celebrates after after firing his team into the lead
Zurich's Blerim Dzemaili celebrates after after firing his team into the lead Photo: Ennio Leanza
4. Zurich's Adrian Guerrero, left, fights for the ball with Hearts' Toby Sibbick.
Zurich's Adrian Guerrero, left, fights for the ball with Hearts' Toby Sibbick. Photo: Ennio Leanza