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 La Coruna 1-0 at home after Knut Rindaroy deflected a low cross into his own net after 34 minutes.
Although the hosts dominated they failed to build on their lead and the visitors struck the post through Lassad in the second half.
On Wednesday, Barcelona thrashed second division Real Betis 5-0 at the Nou Camp and holders Sevilla came back to draw 3-3 at Villarreal. The second legs of all four matches are next week.
REAL PACE
Atletico have not won a city derby since 1999 but grabbed an early goal when Sergio Aguero sprinted away to be brought down by Iker Casillas, only for Uruguay's Forlan to put away the loose ball.
Television replays showed Aguero had started from an offside position.
Real quickly took charge and after Ramos levelled at a corner in the 14th minute they were utterly dominant as Atletico struggling to handle the pace of their attacks.
On the hour mark, Forlan broke the siege to fire against a post after a breakaway from Aguero, but Real edged ahead almost immediately.
The outstanding Ozil set up Ronaldo to score at the backpost and the Germany midfielder then forced the error at the end and reaped the reward himself.41
