Zidane handed a start to Martin Odegaard as the Norwegian made his first league appearance for Los Blancos since 2015 and he impressed against the team he played for on loan last season.
Madrid had the better of an intense and entertaining first half, but the best chance of the opening 45 minutes fell to Alexander Isak just before the break.
The Swede seemed certain to score with a right-footed effort, but Thibaut Courtois stretched out a leg to make a spectacular save from the striker.
Real Sociedad created another fantastic opportunity early in the second half, when Mikel Oyarzabal set up Ander Barrenetxea and the youngster volleyed just wide of the target with only Courtois to beat.
Madrid striker Karim Benzema, who had been off target with a couple of efforts in the first half, was put through by Odegaard later on but saw his effort deflected wide.
Zidane made three changes with 20 minutes left, 20-year-old Marvin among them, and the Nigerian should have hit a shot first time when set up by Vinicius Junior.
The chance was lost and at the other end, Real Sociedad debutant David Silva saw a curler blocked by Raphael Varane with two minutes left.
Both sides attacked in the final stages and in three minutes of added time, but the two teams had to be content with a draw, which leaves Real Sociedad on two points with two games played and Real Madrid on one after their first fixture in the 2020-21 season.