The defeat leaves Real in fifth place, 13 points behind leaders Barcelona, who beat Levante 2-1 on Tuesday.
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Second-placed Atletico Madrid can close the gap on Barcelona to eight points if they beat Levante on Sunday.
Elsewhere, Iker Munian's late goal handed Athletic Bilbao a much-needed victory over Valencia while Levante and Getafe drew 0-0.
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Celta Vigo won 1-0 at Levante to escape the relegation zone for now with its first back-to-back victories of the season.
He also had spells with Spanish side Levante and Sunderland and made over 60 appearances for the Republic of Ireland between 1996 and 2007.
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In Saturday's other matches, Celta Vigo beat Levante 1-0 while Zaragoza moved off the bottom of the table after beating fellow strugglers Mallorca 3-2.
Midfielder Sergio Gonzalez, without a club since being released by Levante in 2011, gave the Catalan team the lead from the penalty spot after three minutes.
In La Liga's other Saturday matches, Valencia played out a 2-2 draw with Levante while the game between bottom-of-the-table Deportivo La Coruna and ninth-place Rayo Vallecano ended goalless.
In Saturday's other fixtures, Malaga are up to second after beating Levante 3-1 while Valencia recorded their first win of the season with a 2-1 win against Celta Vigo.
Real Zaragoza moved out of the bottom three with a 3-0 win that left Rayo Vallecano behind Sevilla on goal difference, while bottom club Mallorca drew 1-1 with midtable Levante.
Elsewhere, Atletico Madrid maintained their seven point lead over their city rivals in second with a comfortable 2-0 win over Levante , though their star striker Falcao limped off with injury.
Levante earlier won 2-1 at home to midtable Valladolid thanks to a 90th-minute own goal from Serbian defender Antonio Rukavina, which put "Los Granotes" behind their Valencian rivals on goal difference.
Real Madrid, meanwhile, came from behind to crush Levante 5-1 as Mesut Ozil scored two late goals after fellow substitute Cristiano Ronaldo also netted -- his 29th in the league, second best behind Messi's 43.
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Two goals in two minutes late in the match from from Ruud Van Nistelrooy and Martin Demichelis ensured Malaga moved three points clear of fifth-placed Levante , who were beaten 2-0 at home by Osasuna.
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