Los Blancos passed up an opportunity to close the eight-point gap to Barcelona with the stalemate.
Real had Sergio Ramos sent off on a frustrating evening where Cristiano Ronaldo and Karim Benzema hit the post.
Earlier in the day league-leaders Barcelona drew 2-2 with Celta Vigo, giving Zidane’s side the perfect opportunity to home-in on top spot.
But they failed to take advantage and remain fourth in the table behind Valencia and Atletico Madrid.
And when asked about his side’s strength in depth, Zidane instead suggested the fault lied with strikers Cristiano Ronaldo and Karim Benzema.
"Real Madrid are eight points behind Barcelona because we don't score enough," Zidane fumed after the game.