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Pasveer saved five penalties and scored one himself to squeeze Ajax past Greek side Panathinaikos in one of the longest penalty shootouts in history.
Olympiakos became the first Greek club to lift a European trophy.
It was just the second major trophy for the Italian side and ended the German champions' remarkable unbeaten streak.
The Italians will take on Bayer Leverkusen, who overcame AS Roma, in their first European final in Dublin on May 22.
Olympique de Marseille claimed a penalty shootout win over Benfica on Thursday to reach the Europa League semi-finals
Bundesliga leaders Bayer Leverkusen secured a last-gasp 2-0 home win over West Ham United.
Sevilla, the undisputed kings of the Europa League, have now won all seven of the finals they have played in the competition, and are well-versed in the drama of the occasion, having seen their opponents score first in the last four finals
The Spaniards will bid for a record-extending seventh Europa League title when they face AS Roma in the final in Budapest on May 31, after Jose Mourinho’s side drew 0-0 at Bayer Leverkusen to secure a 1-0 aggregate victory in their semi-final
Manchester United came from behind to beat Barcelona 2-1 in Thursday's playoff second leg and progress 4-3 on aggregate.
Manchester United won the match against Sheriff Tiraspol, sealing a place in the Europa League knockout rounds.
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