Former Barcelona forward Cristian Tello struck in the 82nd minute as Madrid stayed two points behind its fierce rival, despite having triumphed in last week's El Clasico.
Zidane accepted blame for the result, which came despite Karim Benzema's penalty cancelling out Sidnei's opener in the first half.
"I blame myself for everything. I am responsible," the Madrid manager said. "I will not explain what has happened but it has been our worst game of the season."
With 11 games remaining in LaLiga, Madrid – whose last league title came in 2017 – sits two points adrift of Barca.
While backing his players, Zidane also said they would be made aware of their deficiencies.
"I will defend my players, if I have something to say I will tell them," he said.
"We have lost many balls, which never happens to us and that has been our problem."