Antoine Griezmann settled the derby with a clinical finish in the 53rd minute at Santiago Bernabeu, as Real fell nine points behind La Liga leaders Barcelona, who host Sevilla on Sunday.
Some Madrid fans called for president Florentino Perez to resign during the game, although those chants were swiftly drowned out by whistles from Perez's supporters.
And with hopes of a first Liga title since 2012 seemingly fading away, Zidane conceded his job may come under pressure.
"Next year there might be changes, players and coach. But for now we must keep going as we have something to aim for," Zidane said in his post-match news conference.
"La Liga is not over, but it was difficult before the game. Now even more. But we will continue, we will not throw in this season.
"Today is very hard, a very hard blow, to lose at home to Atletico Madrid. It is difficult. The truth is I am not happy with the game, I cannot be happy with the defeat, with what happened, the game we played.
"Of course it is a difficult moment, it always screws you to lose a derby at home. But we must be professional and think about the next game.
READ MORE | Ronaldo SLAMS Madrid team-mates
"They [Atletico] played on Wednesday [in the Champions League], had less time to recover, and look at the game they played. So it is a mental question - we lacked a bit of everything.
"We were not ready for the game - as we thought we were. We had to run more, against a rival who had played on Wednesday.
"I have the responsibility. The players play, but I have the responsibility. I must look for solutions so it does not happen."
Zidane also explained the half-time substitution of Karim Benzema, who had been ineffective in the opening 45 minutes and was replaced by Borja Mayoral.
"Benzema had a problem, he was feeling pain. He could not sprint, we did not want to take a risk. So he was changed," the coach said.