Twice the Spanish side came from behind to take the tie to extra time thanks to goals from Wissam Ben Yedder and Munir El-Haddadi.
However, after Franco Vazquez had given Sevilla a 3-2 lead on the night, Slavia hit back and Ibrahim Traore’s winner a minute from time sealed a famous win for the Czechs.
Napoli wrapped up a 4-3 aggregate victory over last season’s semi-finalists Salzburg despite a 3-1 second-leg loss in Austria.
Another Serie A giant fell, though, as the highly-rated Luka Jovic scored the only goal of the night and the tie as Eintracht Frankfurt beat Inter Milan at the San Siro.
Russian outfit Krasnodar suffered heartbreak as Goncalo Guedes scored a dramatic injury-time equaliser to send Valencia through 3-2 on aggregate after a 1-1 second-leg draw.
Spanish sides have dominated this competition in recent years and despite Sevilla’s defeat will have two representatives in the last eight as Villarreal beat Zenit St Petersburg 2-1 to seal a resounding 5-2 aggregate victory.
Benfica, runners-up in 2013 and 2014, were also taken to extra-time by Dinamo Zagreb but eventually overturned a 1-0 first leg deficit with a 3-0 win in Lisbon.