Hosting Sociedad took control of the encounter early on and eventually broke the deadlock with 18 minutes on the clock. Former Arsenal striker Carlos Vela was brought down inside the area by Nikolaos Karabelas and the Mexican international dusted himself down to convert the resultant penalty.
The same player went close to putting Levante in front shortly after the restart, but his rising effort from 20 yards out bounced to safety via the crossbar.
Levante's chances of claiming maximum points were then dealt a blow with 58 minutes played as Karabelas was handed his second yellow card of the encounter for another foul on Vela.
Now that they had the numerical advantage, the home side pushed forward in search of a winning goal. Their best chance fell to Vela with 20 minutes remaining, but his header was palmed clear by Keylor Navas.