The Basques had more possession and created more chances than the visitors, who were once again grateful to their goalkeeper Keylor Navas for making a number of crucial saves and keeping a clean sheet.
But La Real have produced better performances this year and cannot feel too large a sense of injustice at only picking up one point.
They remain sixth in the La Liga table, six points behind Athletic and three better off than fifth-placed Villarreal, while Levante drop down to 10th. Les Granotes are unbeaten in their last five league games.
La Real lacked a certain firepower up front until Carlos Vela was brought on as a second-half substitute, and their outstanding performer up until then had been Antoine Griezmann, who met his nemesis in Navas.
The Costa Rica international was called into action first after 14 minutes, repelling a shot from Imanol Agirretxe after a flick on by Griezmann.
Next he made a double save, parrying a ferocious shot by the French winger and then scrambling back to swipe Agirrexte's rebound right on the goal-line.
He had to make two saves in quick succession again just before the break, saving a shot and then a header from the ever-present Griezmann.
Sociedad stopper Claudio Bravo then had to do his bit for his team after spending the majority of the opening 45 minutes as a spectator, and kept out a shot from David Barral following a quick Levante break driven by Pedro Lopez.
The hosts thought Loukas Vyntra should have been shown a second yellow card just three minutes after the break for handling the ball as it was going out of play, but the referee gave him the benefit of doubt.
Vela replaced Castro midway through the second half and almost had a sensational impact.
He ran through the midfield beating two Levante players, played a one-two with Agirretxe and then shot a couple of inches wide of the left-hand post.
The Mexican managed to put the ball in the net a few minutes later by getting on the end of a low cross from Griezmann, but was caught narrowly offside.
Both teams had to endure scrambles in their own areas in the closing minutes, but Bravo and Navas - the latter certainly one of the most in-form goalkeepers in Spain at the moment - would not be beaten.