The two-time Copa winners have been in outstanding form in recent weeks, climbing to sixth in LaLiga, and a sixth straight win in all competitions ended a 32-year wait to return the showpiece.
Imanol Alguacil's side still did not quite hit the heights of their stunning quarter-final triumph at Real Madrid, but inspirational captain Mikel Oyarzabal tucked away his second penalty of the tie four minutes before half-time, and Mirandes could not recover.
Nacho Monreal hung up an early cross and Adnan Januzaj directed a towering header from it just wide, but the game's next effort of note did not follow until Limones parried clear Willian Jose's powerful 20-yard drive with 26 minutes on the clock.
That attempt came amid an improved spell from the visitors, and they had their breakthrough before the break after Mickael Malsa blocked Joseba Zaldua's cross with his arm.
Skipper Oyarzabal kept his nerve from 12 yards, sending Limones the wrong way to stretch La Real's aggregate lead and cancel out their opponents' away goal.
Januzaj went agonisingly close to a second shortly after the restart, but his swirling strike from wide on the right crashed away off the crossbar after beating Limones.
Chances were scarce at the other end, however, and Alex Remiro pounced on Matheus Aias' tame close-range header as Sociedad eased to victory, with a stoppage-time second from substitute Alexander Isak disallowed for offside.
What does it mean? Sociedad set for season of success
La Real have hit form at a great time. They are firmly in the hunt for Champions League qualification through LaLiga due to their winning run, and now a major honour is also in their sights. Sociedad will fancy their chances against either potential final opponent, but a clash with rivals Athletic would be particularly special.
Captain keeps his cool
Each leg of this tie was balanced, yet Oyarzabal stepped up with a key strike from the penalty spot on both occasions. Sociedad needed a goal before half-time and, sure enough, their skipper - the team's chief creative force throughout - dispatched the spot-kick.
Franquesa found wanting
It could quite easily have been Januzaj, not Oyarzabal, who proved Sociedad's semi-final hero. The winger gave left-back Enric Franquesa a torrid time. The gulf between the teams was evident each time the defender was isolated, and better end product from Januzaj might well have led to further goals.
La Real's attention returns to LaLiga and a top-four bid, with champions Barcelona up next on Saturday. Mirandes have a more modest trip to Alcorcon on Sunday.