The champions looked rusty and particularly toothless up front, barely creating any chances of note in either half as Karim Benzema found little support in Brazilian teenage forwards Vinicius Jr. and Rodrygo.
"They're two points dropped. It was hard for us to create chances. We lacked the goal. We did well defensively, but we were missing something in attack," Real Madrid manager Zinedine Zidane said.
Sociedad looked more likely to score in either half, with Sweden forward Alexander Isak forcing Real keeper Thibaut Courtois to save with his feet before flashing a shot narrowly over the bar before halftime.
Lively winger Ander Barrenetxea was inches away from breaking the deadlock early in the second period but fired just wide of the far post.
Benzema missed Real's best chance of the first half when he lost his footing as he rounded goalkeeper Alex Remiro and later saw a goal-bound shot blocked by a dive from Sociedad defender Aritz Elustondo.
Madrid keeper Courtois said he was pleased with his side's performance all in all.
"We're happy with how organised we were and how we didn't allow them many chances but the problem was that we couldn't score, we struggled to complete our final passes," he said.
"We were looking strong even though it was our first game, we played with a high level of intensity although we found it more difficult in the second half but in general we can be happy with how our first game has gone."
Real spent their brief close season moving on a number of unwanted players such as Gareth Bale, James Rodriguez, Achraf Hakimi and Sergio Reguilon but the only new additions were returning loanees Martin Odegaard and defender Alvaro Odriozola.
Odegaard started against Sociedad, where he had a highly successful spell last season, but Real were unable to properly command the middle of the park without Casemiro, who started on the bench before coming on in the second half.
"I felt good, but we didn't get the win. I'm not happy because of that, but it was good to play for Madrid... I understand my teammates but its always a bit difficult at the start. They're players of such a high level," Odegaard said.
Substitute Marvin, 20, could have had a dream Madrid debut when a chance fell his way after coming on but failed to keep his composure.
Sociedad's David Silva made his first La Liga appearance in a decade when he came on as a substitute in the second half.