But the league leaders came out stronger after the break and Toni Kroos levelled in the 52nd minute, shortly after Real captain Sergio Ramos had a goal ruled out for offside.
Ramos then scored from the penalty spot to put Zidane's side in front in the 65th minute after Eden Hazard, making his first outing in almost three months after recovering from an ankle injury, was tripped by Celta keeper Ruben Blanco.
"Dropping two points at home hurts me a lot," Zidane said in the post-match press conference.
"Celta are a good team that don't deserve to be near the bottom of the league, but it's painful to lose the two points after we made such a big effort," he added.
Griezmann to Barca's rescue
Earlier, a slick Antoine Griezmann strike set Barcelona on their way against Getafe to a win 2-1.
Griezmann's finish was perfect as he clipped the ball over onrushing goalkeeper David Soria on 33 minutes. Sergi Roberto doubled the scores six minutes after the opener, but a late rally from Getafe gave Barca a scare.
Meanwhile, Atletico Madrid led twice at the Mestalla before being held to a draw by Valencia.
But, it should be a result that could give Atletico a boost ahead of hosting Liverpool in the first leg of their last-16 tie in the UEFA Champions League.