Granada might have levelled the match after 13 minutes with an effort from Mikel Rico, but his shot from outside the box drifted just wide.
Franco Zuculini might have doubled Zaragoza's lead after 25 minutes, but his effort from outside the box missed the post.
Granada's best chance of the first period fell to Dani Benitez after 38 minutes, but his effort was blocked by the home side's defence, before Rico fired wide just before the break.
It was Zaragoza who started the second period the better and Dujmovic twice came close after 48 minutes.
Zaragoza continued to carve out openings, and Zuculini tested Julio Cesar just before the 60-minute mark in the match.
Borja Gomez came close for the away side after 61 minutes, but his header from close range missed the target.
Benitez was next to come close for Granada after 81 minutes, but his effort from outside the box smashed off the post.
Zaragoza held on in the final moments, however, to secure all three points.