The last time Real drew four games in all competitions was in March 2006, when the club were reeling from on-field and institutional crises and president Florentino Perez had recently resigned.
Although Real remain unbeaten in the league, they are without a win in their last three games and are on 15 points from seven games, the number Zidane's predecessor Rafael Benitez had earned at the same stage last season.
The Spaniard was sacked three months later.
"We are not in crisis, we're only in October. What we need to do is improve, we can't lose our minds over this," Zidane told a news conference on Sunday.
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"I'm not going to go crazy and neither are my players, all I can see is that we need to work harder and be more intense at the start of games."
Real fell behind to Fran Rico's header in the sixth minute after a lapse in concentration at the back and a mistake from goalkeeper Keylor Navas, who let the ball slip through his hands.
The Costa Rica international returned to the side only last Tuesday against Borussia Dortmund after more than four months out with an Achilles injury and was also at fault for one of the goals in the 2-2 draw in Germany.
Gareth Bale equalised later in the first half but Real, missing Luka Modric, Casemiro and James Rodriguez through injury, struggled to create many chances against the Basques, although Bale hit the post after the break.
Real's next game is away to Real Betis in a fortnight's time, after the international break.
"We leave tonight with a bad taste in the mouth. We let our opponents play in the first half and we knew that they could score like that and we didn't deal with it," added the coach, who lead Real to their 11th European Cup last season and last month saw them match the La Liga record of 16 straight wins.
"We can't blame fitness issues, we just need to work with more intensity, starting with me. When the players return from international duty we have to turn this around, we can't continue like this."
(Editing by Clare Fallon)
