Captain Mikel Oyarzabal netted the winner by sending keeper Unai Simon the wrong way from the penalty spot in the 63rd minute after strike partner Portu was felled in the area by Athletic defender Inigo Martinez.
Martinez, a boyhood Sociedad player who left for their rivals for €32 million in a controversial transfer in 2018, was initially shown a red card for the foul but the referee downgraded the punishment to a yellow card after a VAR review.
Sociedad dominated possession and Athletic failed to test their opponents on the counter-attack, still offering little in attack even as they chased the game deep into the second half.
"It's a beautiful day on which so many things come to your mind. Giving this victory to our fans was the best thing we could do," said match-winner Oyarzabal.
"There are many people who would like to have been here and there are many friends and family who left us in the worst possible way. This is for everyone, we felt their presence. There's nothing like this feeling."
Athletic's Oscar de Marcos described the defeat as "a massive blow."
"It's very tough for us, this was a match we were so excited about and thought we could win," he said. "It's very painful whenever you lose a final, regardless of who you lose to. It's a very sad day for us."
Sociedad's vice-captain Asier Illarramendi lifted the trophy next to Oyarzabal after being forced to miss the final due to a thigh injury sustained in Friday's training session, the latest setback in a series of serious injuries which led to him spending 18 months on the sidelines.
The match was postponed by a year at the request of both clubs to ensure fans could attend, but continued coronavirus restrictions in Spain meant the showpiece in Seville's La Cartuja stadium took place without any supporters.
Prior to the match, Athletic fans clashed with police in Bilbao ahead of the final after officers tried to prevent people from gathering in large numbers in defiance of coronavirus restrictions.
Video footage showed fans setting rubbish containers on fire while others hurled glass bottles and other debris at officers close to the team's San Mames stadium, where hundreds of fans congregated before the match, which was held in Seville.
Athletic have now lost their last five Copa finals, but have another shot at the trophy when they play Barcelona in this season's showpiece back at La Cartuja on April 17.