After the game finished 2-2 following extra time, Sociedad goalkeeper Unai Marrero saved shots by Alexander Sorloth and Julian Alvarez, before Pablo Marín converted the last kick to clinch the shootout 4-3 on Saturday (Sunday AEST).
"I knew that if we got to penalties, I believed in myself, my teammates believed in me, our fans believed in me," the 24-year-old Marrero said before the winners received the trophy and medals from Spanish King Felipe VI.
"I still can't believe it. The boy who dreamed for this since he was young has fulfilled his dream."
The New Jersey-born Matarazzo took over Sociedad in December when they were two points above the relegation zone in the Spanish league.
Four months later, the Basque Country club have won the cup and are in the upper half of the La Liga table.
Sociedad needed 14 seconds and just three players to touch the ball directly after kickoff to take the lead through Ander Barrenetxea. Opta Statistics said it was the fastest ever in a Spanish cup final.
Ademola Lookman equalised for Diego Simeone's side with a goal in the 19th minute, but a penalty by goalkeeper Juan Musso allowed Mikel Oyarzabal to restore Sociedad's advantage in first-half injury time.
Alvarez stroked a shot into the right corner to make it 2-2 in the 83rd minute.
In the shootout it was Marrero's turn to shine as he guessed right on the first two penalties by Atletico's strikers.
"The team did all it could to turn this around, we were behind right from the start and did what we could," Atletico's veteran midfielder Koke Resurreccion said.
"It is a difficult night. Thanks to all our fans who travelled with us."
Atletico veteran Antoine Griezmann will now need to help get his side past Arsenal in the European semi-finals to have another shot at finishing his final season in Spain with a title before the club's all-time leading scorer joins Orlando City in the MLS next season.
"We want to win the Champions League," Koke said.
"But tonight is a hard night. We will have time to think about the Champions League."
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