Oyarzabal’s first penalty came in the 30th minute when, following consultation with the VAR, the referee decided Inigo Martinez’s challenge on Luca Sangalli inside the area was illegal.
Athletic were behind for just two minutes, however, with Iker Muniain drawing them level after finishing off a move he had started.
The diminutive striker wriggled away from three opponents on the halfway line and was then in the right place to stab home after Sociedad had made a mess of dealing with Markel Susaeta’s right-wing cross.
It would be Sociedad who would claim derby bragging rights, though, as they took advantage of two defensive mistakes in the second half to seal the win.
Sangalli made it 2-1 in the 47th minute with a clinical finish following good work by Igor Zubeldia, who intercepted a loose pass before charging forward and crossing for his colleague.
Then, with 16 minutes to go, Sociedad were awarded a second penalty when Jon Bautista got to Alvarez Yeray’s under-cooked back pass ahead of Unai Simon before being sent flying by the Athletic goalkeeper.
The referee immediately pointed to the spot and Oyarzabal sent Simon the wrong way from 12 yards as Sociedad climbed up to eighth in LaLiga.