Despite still missing key players due to injuries, Real Madrid hit the post twice in the first half, but the visitors took the lead in the 55th minute with a towering header from forward Portu.
The hosts passed up many opportunities to articulate a swift equalizer and Real Sociedad were heading home with the three points until substitute Vinicius Junior came off the bench to snatch a 1-1 draw.
Los Blancos had the chance to move within three points of leaders Atletico Madrid ahead of the Madrid Derby on Sunday, but the draw left them third on 53 points from 25 games, level with second-placed Barcelona and five adrift of Diego Simeone's men.
"I changed it because I wasn't happy with our pressing," Zidane told Marca after the game.
"If you've asked me and that's now three times, then yes, maybe," Zidane replied when asked whether the tactical shift had hurt his side.
"We had to change things because, after an hour, the team was a bit tired. Sometimes we have to change things.
"In the end, you try to find things to change the dynamic."
Real Madrid travel to the Wanda Metropolitano on Sunday, and Zidane says he is happy with the form of his team.
"We have to remain calm," Zidane said. "We had a good game and we had three or four chances [to score].
"Patience, calm and rest. We were up against a team that creates chances against you.
"We have to carry on.
"It doesn't change the way we go into the derby. We have to go there to have a great game."
