The Catalans played Wednesday's semifinal in Cordoba without the services of captain Lionel Messi, who was ruled out with an undisclosed injury that saw him subbed out of last week's league win over Granada.
Barca advance to the final of the tournament -- which features the winners and runners-up of last season's La Liga and the Copa del Rey -- with the other semifinal set for Thursday against Real Madrid and Athletic Bilbao in Malaga. The final will be held at La Cartuja Stadium in Seville on Sunday.
But La Real equalised the affair in the 51st minute when VAR confirmed the ball hit De Jong's arm on a cross into the box, with Mikel Oyarzabal converting the successful kick from the spot.
Both teams went back and forth with several good looks for the duration, but neither side were able to find the second goal before penalties decided the match.
Both De Jong and Griezmann missed their attempts in the penalty shootout but Ousmane Dembele, Miralem Pjanic, and Ricard Puig were successful in their chances. Mikel Merino and Adnan Januzaj converted for the Basque side.
"I had four names down to take penalties, I asked who took the fifth and Riqui said he would. Perfect," said Barca coach Ronald Koeman.
"It was a great game, both sides played really well, it was really intense and both teams had chances to win the game."
Puig is one of the most hyped players to emerge from Barca's youth academy in recent years but despite remarkable technical ability, he has barely featured for the first team since Ronald Koeman took charge this season.
Even with numerous injuries, Puig is yet to earn a league start under the Dutch coach, who has at times got frustrated by constant questions from local media about why he has not given the youngster more of a chance.
"The coach said he had four takers and he needed a fifth and I was the first one to say I wanted to take it," Puig said.
"They had missed many penalties, so had we, Antoine missed one that could have been decisive but when I picked the ball up I knew it was going in the net and I struck it with real conviction."
"I never lose my smile, I'm a happy boy even though I'm not playing much and if Ronald gives me a the odd minute here or there I'll make the most of it," added the midfielder.
"If not I'll keep working, I'm never going to throw in the towel."
Oyarzabal said he was proud of Sociedad's efforts over the 120 minutes but admitted losing on penalties "hurt even more".