Not since 2002 had El Classico finished 0-0 and the result means that Barcelona maintain their advantage at the top of the La Liga table on goal difference alone.
However, it was Zidane that was the happier of the two coaches after the game and says his players were right on the money from the first whistle until the last.
"I'm very happy," Zidane told a press conference. "I think how we went out on the pitch, how we played from the first minute to the last, I think we played an incredible match.
"We were just missing a goal. We deserved a bit more today, but that's football. When you have opportunities to score you have to convert them and what we were missing was a bit of clarity in the final touch."
Chances were few and far between during the game, with Barca's best opportunity falling to Jordi Alba, who could only fire wide from a good position in the first half.
For Real, Gareth Bale, who was restored to the starting line-up, thought he had won it in the second half but his goal was ruled out by VAR for offside.
Zidane has now urged his side to push on, while Barca boss Ernesto Valverde admitted after the game that his side found it difficult to impose themselves on the game against a well organised Real side.
"It was a very disputed match in which there were phases for both teams," Valverde said. "They squeezed us high and created problems with high balls. That's where we've suffered."