Having beaten the Danes 2-1 in their own backyard seven days ago, Sociedad were expected to start on top.
And they did just that inside five minutes, shaking off any ill effects from a 3-0 weekend loss to Betis when Ousmane Diao under-hit a pass to Mads Bech Sorensen, who lost his footing in the box and allowed Brais Mendez to steal the ball and net the opener - just as he did in the first leg.
The Basque side only sustained the pressure and doubled their lead on 18’, when a killer pass from Mikel Oyarzabal to Ander Barrenetxea allowed him to roll the ball across the area for Luka Sucic to tap in.
It looked as though Midtjylland were in for a long night, but from out of nowhere, they pulled a goal back from the spot after Barrenetxea got the slightest of touches on Denil Castillo inside the box.
Theatrical his fall might have been, but Adam Buksa took full advantage firing the ball into the bottom-right corner after sending Alex Remiro the wrong way.
Unthinkably, they were level on the night almost a quarter-hour later, as Mendez gave the ball away to Pedro Bravo, and he found Dario Osorio via Buksa’s flick-on before the Chilean smashed the ball in at the near post with the outside of his boot.
Sociedad roared back and there was time enough for them to retake the lead just before half-time. Takefusa Kubo’s corner was only half-cleated to the edge of the box, and after taking one touch, Sucic blasted beyond Lossl to restore Sociedad’s lead.
While the first half was always going to be a tough act to follow for the neutral, Aritz Elustondo should certainly have done better when the ball fell to him on the edge of the box, only for the defender to send the ball marginally wide of the right-hand post.
There was some drama at the three-quarter mark though, when Dani Silva overzealously put his studs into the ribcage of Igor Zubeldia when challenging for the ball, in retaliation for an earlier foul unto him.
While that wasn’t the decisive coup de grace, the penalty Sociedad won a minute later - when Juninho barged over Oyarzabal inside the box - certainly was.
Oyarzabal himself stepped up and fired into the far-left corner beyond Lossl’s despairing lunge.
Midtjylland knew the game was up, and there was more pain still to come, as Arsen Zakharyan crossed in from the left flank for fellow substitute Orri Oskarsson to steer home at point-blank range for his fourth goal in Europe.
In the end, Sociedad’s first home win in a post-xmas European knockout tie this century was a cruise.
As a ‘reward’, however, they will next face either of Premier League duo Tottenham or Manchester United - both of which are still clear Europa League outright favourites with many bookmakers despite having poor league campaigns to date.