The match was tetchy with referee Anthony Taylor dishing out 14 yellow cards, the most ever in a Europa League game, and playing almost 30 minutes of stoppage time in total.
Their run to the final saw them beat PSV Eindhoven, Fenerbahce and Manchester United before downing Juventus in the semis.
It was the third consecutive Europa League final that ended in a penalty shootout.
The win means Sevilla will compete in next season's Champions League despite finishing outside the top four in the LaLiga.
Mourinho had never lost a European final before Wednesday, having last year led Roma to the inaugural Europa Conference League title, becoming the first coach to win all the European trophies.
Persons:
John Sibley, Montiel, Mourinho, Jose Mourinho, Sevilla, Gonzalo Montiel, Rui Patricio, Yassine Bounou, Gianluca Mancini, Roger Ibanez, Anthony Taylor dishing, Paulo Dybala, Mancini, Roma, Chris, Lucas Ocampos, Movistar, Fernando Kallas, Toby Davis
Organizations:
Soccer, Europa, Sevilla, AS Roma, Puskas Arena, Europa League REUTERS, BUDAPEST, Europa League thoroughbreds Sevilla, Roma, France, Europa League, Champions League, PSV Eindhoven, Fenerbahce, Manchester United, Juventus, League, inaugural Europa Conference League, Thomson
Locations:
Budapest, Hungary, Sevilla, Argentina, Spanish, Portuguese, Madrid