Athletic goalkeeper Julen Agirrezabala saved a penalty by Manuel Morlanes, while Mallorca's Nemanja Radonjic shot high. Raul García, Iker Muniain, Mikel Vesga, and Alejandro Berenguer - all substitutes - converted their spot kicks past Mallorca goalkeeper Dominik Greif.
Extra time in the final ended still locked at 1-1 after Dani Rodríguez gave Mallorca the lead in the 21st minute and Athletic's Oihan Sancet levelled in the 50th of Saturday's decider at the packed 57,000-seat La Cartuja Stadium.
Muniain received the trophy from Spanish King Felipe VI before lifting it high for his team and fans to celebrate.
Athletic won their 23rd and last Copa in 1984. Since then they had lost six straight finals, including in 2020 and 2021. To win this one, they had to first beat both Barcelona and Atletico Madrid in the quarter-finals and semi-finals respectively.
"This is incredible. Our fans have always been behind us and now it is time to enjoy this," Athletic forward Nico Williams said. "We have been fighting for this for a long time."
Only Barcelona and its 31 Copas has more than Athletic, who excels at the tournament despite fielding players only from or near Spain's northern Basque Country region.
NBA great Steve Nash, a minority owner of Mallorca, was at the game, as well as Rafael Nadal, a native of the island of Mallorca.
Even though Athletic are fifth in the Spanish league and Mallorca down in 15th place, there was little difference between them in the final.
Javier Aguirre, Mallorca's journeyman coach, succeeded in nullifying the more explosive attack of Ernesto Valverde's Athletic.
Athletic were unable to stop Mallorca from getting three consecutive shots in their area after a corner kick before Rodríguez opened the scoring. The midfielder curled a shot into the corner of the net after Antonio Raíllo controlled a rebound and laid the ball off.
Athletic took time to finally click in attack, and when it did most of the damage was done by Williams.
The Spain winger had a goal waived off for offside before he shot into the side netting in first-half injury time. Grief also saved an early shot by Iñigo Galarreta, while Gorka Guruzeta shot wide.
Canada forward Cyle Larin had an opportunity to double Mallorca's lead seconds after halftime, but Agirrezabala saved his effort on the break.
Williams helped produce Sancet's equaliser when he recovered the ball and fed a throughball to the midfielder to beat Greif.
But Sancet's goal did not fluster Mallorca's defense, which again clamped down after Aguirre refreshed his side with substitutes.
After a long stalemate without any clear chance, Williams had a point-blank shot that grazed Mallorca defender Pablo Maffeo and hit the post.
Vedat Muriqi then had a header parried by Agirrezabala on the other end as the game finally came alive before going to the shootout.
Muriqi and Antonio Sánchez converted their kicks in the shootout for Mallorca.
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