The Portuguese striker scored his first goal since joining from Sporting Lisbon with a stunning 11th minute overhead kick from the edge of the area that flew past Claudio Bravo in the Sociedad goal.
Both sides had chances as the game flowed from end to end in the first-half but Postiga scored the telling second goal just three minutes into the second period when he collected Leonardo Ponzio's fine through-ball and slotted past Bravo from close range.
Zaragoza's results had improved slightly before the international break after a poor start to the campaign but this was by far their best performance of the season to date as they moved the ball quickly - and in Luis Garcia and Postiga they had two willing and intelligent runners up front.
Postiga's first goal was a display of fantastic technique as he contorted his body to connect with Efrain Juarez's cross on the volley and fire it into the top corner from 15 yards.
The early goal seemed to give the hosts a great deal of confidence and they were unlucky not to get a second when Javier Paredes' free-kick can back off the base of the post and Indigo Martinez made a fine saving challenge to stop Postiga tapping home the rebound.
However Sociedad ended the half the better and twice deflected David Zurutuza shots nearly had them on level terms.
First, his drive from outside the area dribbled just past the post - and from the resulting corner a more potent shot was deflected into the ground and palmed away by Roberto.
The home side started the second period as they had the first and extended their lead on 48 minutes when Ponzio evaded two weak Sociedad challenges with ease before slipping the ball through to Postiga, who took his time before dinking the ball over Bravo.
Imanol Agirretxe nearly pulled a goal back seven minutes later but his shot across Roberto was well held by the keeper.
Thereafter Zaragoza sat back and allowed the visitors to have more possession - but rarely did the visitors look like breaking the hosts down.
Their last chance came from a rare error by Roberto as he misjudged Carlos Martinez's cross and Antoine Griezmann's header nearly found the net.
However, there was to be no equaliser for Philippe Montanier's men, who slip to 12th in the table as a result of Sunday's defeat.