La Real stunned Real Madrid at the Santiago Bernabeu in the previous round, but Mirandes had knocked out three LaLiga teams en route to the last four and caused concern for top-flight opposition once more.
Captain Mikel Oyarzabal dispatched a ninth-minute spot-kick for Sociedad, yet they were pegged back by Matheus Aias and scarcely deserved the second Martin Odegaard lashed in to restore their lead before the interval.
The second period brought marginal improvement in the home ranks but no further goals, leaving the tie in the balance ahead of the return fixture next month.
2-1 - Real Madrid Castilla were last team from Segunda División to play a Copa's Final (1980). In that edition, "the Whites" eliminated Real Sociedad after losing 1-2 in the first leg in San Sebastian (Quarter-Finals). Déjàvu#RealSociedadMirandes #CopadelRey2020 pic.twitter.com/BxmxCp555f— OptaJose (@OptaJose) February 13, 2020
Sociedad's opener came only after Alex Remiro was forced into a smart, low stop from Martin Merquelanz with just five minutes on the clock.
Portu earned a penalty as Odei Onaindia dallied in the area and allowed the home winger to get between man and ball before going down, and Oyarzabal coolly converted from 12 yards.
Mirandes deservedly levelled six minutes before half-time, as Mickael Malsa pounced on some careless play 20 yards from goal and fed Matheus Aias, the striker twisting to send in a low shot that deflected beyond Remiro.
But Sociedad responded swiftly. Limones spilled Odegaard's powerful drive but recovered well to block Portu's follow-up, only for the Real Madrid loanee to rifle in a second rebound.
Matheus Aias blazed over from close range heading into the break, while there were big saves at either end following the restart, Limones denying Portu before Remiro parried an Antonio Sanchez strike.
The two goalkeepers were largely underworked thereafter until Limones superbly turned away a header from one of his own defenders in the fourth minute of stoppage time, teeing up an intriguing second leg at Estadio Municipal de Anduva.
What does it mean? Sociedad warned
Mirandes' earlier exploits in this competition meant La Real had no excuse as they faced this stern examination, surely well aware of the threat their opposition posed. The LaLiga side still struggled to get to grips with Merquelanz and Alvaro Rey, though, and must be better in the return leg to get the job done.
Martin up for the cup
Odegaard is enjoying a fine season in San Sebastian, and he has saved some of his best performances for this cup run. His third goal in the competition this term came at a crucial time, while a gorgeous pass early in the second half set Portu through, the busy wide man unable to beat Limones for a two-goal lead that might have put the tie beyond Mirandes.
Lacking in key areas
On the balance of play - particularly in the first half - Mirandes were good value for at least a draw from this first leg, but shortcomings at either end of the pitch proved costly. Onaindia paid the price as he reacted slowly to a ball into the box and felled Portu, while Matheus Aias, despite scoring a fortunate equaliser, missed one big opportunity.
Focus returns to league action for the coming weeks, with La Real visiting Eibar on Sunday as Mirandes host Albacete. The two sides reconvene for the Copa second leg at Mirandes on March 4.