Raul Garcia's first-half double was enough to knock the holder out and end its nine-match unbeaten run at La Rosaleda.
Karim Benzema's 13th goal of the season gave Los Blancos hope after Marco Asensio hit the woodwork twice, but the Basque side held on to secure only a second win over the LaLiga champion in 23 matches and a first victory under new coach Marcelino.
Madrid boss Zidane felt his side fell short of the high standards he expects in the first half, but he will not be dwelling on the setback.
"Our first half was difficult, we didn't get into the game well, they scored two goals," Zidane said. "They had two chances and two goals. When you are two goals down it is always difficult.
"Then we had a better second half, we created chances, we had several, scored a goal, but could not score a second."
"It is not a failure. Failure is not trying, not giving everything on the field. Life is like that, you can't always win. What we always try to do is win, but you can't always."
Madrid had not lost since 1 December and Zidane expects his side, which has a weekend off before facing Alcoyano in the Copa del Rey, to respond after missing out on another trophy."We had a good run and now it's two draws and one loss [in their past four games]. We have to keep working to change the dynamics quickly.
"The outside will be the same as always, what we can do is keep working and move this forward as a team," he said.
Eden Hazard again failed to show what he is capable of, hitting the target only once from three attempts and making just one key pass.
The Belgium forward has scored just two goals and not provided an assist in an injury-hit campaign, but Zidane remains confident he will come good.
Zidane said: "He must regain confidence, play a good game, score a goal, something different. We know the player he is and we have to be patient, but he is working. We are with him, we have to be patient."
Athletic and Barca do battle for the trophy at Estadio de La Cartuja on Monday (AEDT).