Real Sociedad's first penalty disappointment took place in first-half stoppage time when Brais Mendez saw his spot-kick saved by Mallorca's Slovakian goalkeeper Dominik Greif.
Giovanni Gonzalez went on to give the visitors the lead in the opening stages of the second period, before Mikel Oyarzabal came off the bench to net a 71st-minute equaliser, sending the semi-final tie to extra-time and penalties.
Unfortunately for the hosts, Oyarzabal was punished for failing to convert the first spot-kick as Mallorca kept their composure to score all five penalties and book their place in the final for the first time since winning the competition in 2002-03.
The two sides locked horns for the third time this month after the goalless first leg was followed by a 2-1 win for Real Sociedad in a league meeting on February 18.
The first chance of a cagey half did not arrive until the 29th minute when Arsen Zakharyan saw his ambitious effort comfortably saved by Greif.
Around 10 minutes later, Mallorca's Abdon Prats spurned the chance to hit the target as he dragged his strike harmlessly wide from the edge of the box.
Although the majority of the first period passed by with minimum fuss, Real Sociedad were presented with a chance to take the lead on the stroke of half time when Antonio Raillo was penalised for handling the ball in the box.
The Mallorca captain would have been grateful to see Greif make an important save with his leg to deny Mendez from the penalty spot.
From nearly going behind, the visitors took the lead through Gonzalez inside the opening five minutes of the second period.
The 29-year-old arrived at the far post to meet Jaume Costa's lofted cross and head the ball past the helpless Alex Remiro.
Real Sociedad boss Imanol Alguacil decided the time was right to make his second change of the game as he summoned the attacking qualities of Oyarzabal from the bench.
The Spain international required less than 10 minutes to make a crucial impact, slotting the ball into the bottom corner after being teed up by Mendez.
The Basque side quickly went in search of a winner, with Sheraldo Becker's curling effort drawing a diving stop from Greif.
La Real continued to pile on the pressure in extra-time, as Kieran Tierney saw his fierce shot blocked on the line by Costa.
With no goalline technology, VAR was required to rule that Tierney's effort had failed to cross the line, denying the Scotsman his first goal for the club.
After 210 minutes of action across the two legs, penalties were ultimately required to determine a winner at the Reale Arena.
Nine out of the 10 penalties were scored, but crucially, Greif made a decisive save to deny Oyarzbal's in the shootout's first spot-kick, securing Javier Aguirre's side a place in the final against either Athletic Bilbao or Atletico Madrid.
While Mallorca celebrate a memorable evening, Real Sociedad are left to reflect on a poor run of form that has seen them win just one of their last nine matches.