Valencia had two players sent off as they fell to a shock 2-1 defeat at lowly Las Palmas, while Atletico Madrid gave leaders Barcelona the chance to pull 11 points clear after only drawing 1-1 with Girona.
A 26th-minute penalty from Manu Trigueros and Real youth product Denis Cheryshev's first league goal in nearly two years saw Javier Calleja's side, who beat Real at the Santiago Bernabeu for the first time ever last week, move up to 34 points, two ahead of Real.
The last time Real finished outside the top four was in 2000, when they placed fifth but won the Champions League -- a year before coach Zidane had even signed for the club as a player.
Madrid do have two games in hand on the teams around them, though, and host Deportivo on Sunday.
Cheryshev's tap-in, the first goal in La Liga for the injury-plagued attacker since March 2016, sealed the deal against Levante, who pulled one back with a stoppage-time penalty from Roger.
- Atletico and Valencia slip -
==============================
Antoine Griezmann's first goal in six league games wasn't enough for Atletico as they drew 1-1 with Girona and gave Barcelona a chance to extend their lead at the top.
The draw at the Wanda Metropolitano means Atletico are second and eight points behind Barca, who can almost end discussion over the league title with a win at Real Betis on Sunday.
Valencia looked set for a routine win when Santi Mina headed home the opener from a corner after just five minutes, but Jonathan Viera smashed home Calleri's knock-down 15 minutes later.
Calleri slotted home the penalty that sealed the three points as Gabriel strode off to the showers.
Costa made his mark again on Saturday, with his second assist since returning from Chelsea adding to his three goals in five games in all competitions.
He lay down in a heap after being clattered by Girona goalkeeper Bono as he headed on for Griezmann, but it was the shot-stopper who came off worse and had to be substituted at half-time.
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