"It's not a failure," Zidane said after the game. "A failure is not to give your all on the pitch and not to try. The team tried but we couldn't get the equaliser. Life is like this. You can't always win."
Zidane admitted his players underperformed in the opening half. Raul Garcia struck twice before the interval to give the Basque outfit a 2-0 half-time lead.
However, the French tactician said he was proud of his team's reaction in the second half. Marco Asensio came close to scoring but was denied twice by the woodwork before teammate Karim Benzema reduced the arrears in the 73rd minute.
"We didn't start the game well," Zidane said. "They had two chances and scored two goals. When you are 2-0 down it's always difficult. The second half was totally different, we created lots of chances, hit the woodwork twice."
Zidane, whose side were held to a goalless draw at Osasuna on Saturday in La Liga, believes his team must improve.
"We need to turn the page and continue to work," he said. "We can't go mad, we just have to keep working. When things don't go well you have to work and insist on that. Now we need to rest and think about our next game."
Zidane also expressed confidence in Eden Hazard despite his continued struggles.
The Belgian, Real's record signing, has only scored two goals in all competitions this season and offered little in attack.
"He needs to recover confidence, play a great game or score a goal, just to do something different. We know what a great player he is and we need to be patient with him as he is working hard," Zidane said.
"We are behind him and we must give him time. The fans want to see the best version of Eden so we must be patient. We're sticking with him and he really wants to step up. But today it wasn't just Eden, we're all in this together."
Real Madrid next visit Spanish third-division side Alcoyano in the Copa del Rey round-of-32 on Jan. 20.