In his 15th match for Sociedad, Finnbogason hit the back of the net in the 27th and 61st minutes to give the Basque club a 2-0 aggregate victory over Real Oviedo.
It was a good day for La Liga clubs with six teams from Spain's top tier of football advancing to the round of 16, although Eibar and Cordoba were eliminated.
Having failed to win in three straight games in all competitions, including a scoreless draw at Oviedo in the first leg, Sociedad bounced back at home in the Copa del Rey for David Moyes' second win since taking over as coach.
Finnbogason, who joined Sociedad from Heerenveen before the season, started just his eighth match and opened the scoring when a long pass forward picked out Pablo Hervias in the box and he squared the ball for the Icelander to tap into an empty net.
The 25-year-old striker wrapped up the match just after the hour mark when Oviedo Gorka Magunazelaia could only parry goalkeeper Aritz Elustondo's thumping drive from outside the area, and Finnbogason was on hand to volley the ball home.
Villarreal, who sit sixth in La Liga, notched their fifth consecutive victory in all competitions, cruising past Cadiz 3-0 to complete a 5-1 aggregate triumph.
Gerard Moreno hit a second-half brace for the home side after Trigueros had given Villarreal the lead from the spot in the 55th minute.
In the two all-La Liga ties completed on Wednesday, Getafe progressed at the expense of Eibar, while Granada eliminated Cordoba.
Getafe defeated Eibar 2-1 thanks to two stunning goals as they ran out 5-1 winners on aggregate.
Carlos Vigaray drove through the middle of Eibar's defence before tucking away the opening goal in the 22nd minute, while Diego Castro got the fans on their feet with a brilliant curling effort from the top corner of the box 11 minutes later.
In Cordoba, Granada had to come from behind to secure a 1-1 draw, which saw them progress 2-1 on aggregate.
In the other ties, Levante drew 0-0 with Albacete to advance on away goals after the first leg ended 1-1, while Espanyol defeated Deportivo Alaves 1-0 to triumph 3-0 over two legs.
