With just one win under his belt since joining the club and after witnessing his side only secure a draw against the Spanish third-tier outfit during the first leg, David Moyes was under pressure to deliver victory at the Anoeta Stadium.
Moments later, Finnbogason was denied again from close range but shortly before the half-hour mark, he netted his first Sociedad goal when he converted from two yards after some fine approach work from Pablo Hervias.
As the opening half drew to a close, the visitors were being restricted to shots from distance, with only David Generelo recording an effort on goal before the break.
However, when the teams returned for the second half, the tide appeared to have turned as Eneko and Jonathan Vila forced saves out of Geronimo Rulli, but soon after the hour mark, Sociedad doubled their advantage.
Finnbogason, who had joined Sociedad from Heerenveen during the summer, fired the ball into the net from inside the six-yard box to all but seal his team's spot in the last 16.
Josep Sene had a free kick saved while Sergio Garcia's shot across goal evaded the far post as Oviedo threatened to halve the deficit, but Sociedad survived before being denied a third goal after Pardo's shot was prevented from finding the top corner.
Oviedo fought on until the final whistle but after the chances dried up, Sociedad were able to comfortably seal their place in the last 16.