Atletico took a seventh-minute lead through Diego Forlan only for Sergio Ramos to level with a header in the 14th as Real dominated in an electric atmosphere at the Bernabeu.
Cristiano Ronaldo put the hosts ahead in the 61st and just when Atletico looked like they might have weathered the storm their defenders gifted Real a third in the 90th.
Luis Felipe stuck out a leg to divert the ball to unsuspecting team-mate Alvaro Dominguez who stumbled backwards allowing it to run between his legs, and Ozil stepped in to score past a motionless David de Gea.
"At 2-1 it was a good result. 3-1 is tougher to come back from but not impossible," Atletico coach Quique Sanchez Flores, who led them to the final last year, told reporters.
"After the players' efforts, to give away such an absurd third goal is a double blow to morale."
Earlier, Almeria put their poor league form behind them to beat Deportivo Coruna 1-0 at home after Knut Rindaroy deflected a low cross into his own net after 34 minutes.
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