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Atletico won 1-0 on the night, with the Croat firing in low from inside the area after just two minutes.
Raul Garcia also went close for the league champions, who had coach Diego Simeone sent off for patting the head of the fourth official.
James Rodriguez had Real's best chances with two shots that flew past the post.
The Colombia World Cup star, who scored in the first leg, also had an effort pushed away by keeper Miguel Angel Moya and headed wide.
Mario Mandzukic signed for Atletico from Bayern Munich for about £20m in the summer
The £71m signing from Monaco was one of Real's better performers inside the Vicente Calderon stadium, with Gareth Bale, Toni Kroos and Karim Benzema all disappointing. And Cristiano Ronaldo barely threatened when he came on in the second half.
As for Atletico, they resembled the battle-hardened side who won the title much more than the team that drew 1-1 against Real in Tuesday's first leg.
Their coach also picked up where he left off in the Champions League final by being sent to the stands by referee David Fernandez Borlaban.
Simeone was seen protesting that right-back Juanfran had not been allowed back onto the field quickly enough after receiving treatment for a facial injury before he patted the fourth official on the head, which led to his dismissal.
The match on the whole was a more entertaining affair and was given an early jolt when Croatia striker Mandzukic scored moments after the kick off.
Moya, who replaced Thibaut Courtois in the summer, launched the ball downfield which found its way to Antoine Griezmann. The France international, acquired from Real Sociedad during pre-season, found Mandzukic who ran in on goal and drove his shot past Iker Casillas.
Real looked for an quick response with Rodriguez leading the fight. The midfielder was first denied by Moya, before twice shooting wide and directing his header off target.
That was the sum of Real's best efforts on the night, with Ronaldo, who started on the bench because of a back injury, barely troubling Moya with a 25-yard strike.
Real also ended the night with 10 men when midfielder Luka Modric was shown a second yellow for clattering into Cristian Rodriguez in stoppage time.
Atletico came close to adding more, with Raul Garcia - arguably the man of the match - going close on three occasions, the best of which was a header that came off the bar.
Simeone, who spent much of the match marching back and forth in the stands, has now guided the Rojiblancos to Spanish league title, the Europa League, Copa del Rey and Uefa Super Cup during his three years in charge.
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