Madrid fell a goal behind before the break when Bruno Soriano converted a penalty to open the scoring at Santiago Bernabeu, before Sergio Ramos eventually restored parity after the interval as the hosts stepped up their game.
Los Blancos failed to net the winner, though, as their winning streak in the league ended at 16 successive matches.
And Zidane has stressed they must improve the way they start games.
"We managed to change things around after the break, but the first half was not good enough. That is not the first time this has happened. We will try to work on this," the Frenchman said at a news conference.
"We will now focus on our next game. We have to change some things, especially the way we start. We like to put the pressure on our opponents right from the start, but we were unable to do that. But we cannot always play well and be at our best from the start. Things like this happen.
"It always costs energy when you have to chase the game. You make life difficult for the opponent if you go a goal up. We have to be more focused from the start. We lacked intensity in the first half.
"But the second half was phenomenal. The second half was spectacular. Mistakes can happen in football. You cannot always secure the win in the final minutes. But I am happy with the team. We got a point in the end and I am happy with our second half."