In a bad-tempered match at a rowdy Nou Camp stadium in Barcelona, winger Capel pounced on a loose ball on the edge of the penalty area in the fifth minute and smashed a left-foot shot past David De Gea.
Atletico forwards Diego Forlan and Sergio Aguero went close to equalising before Navas skipped past several lunging tackles, took the ball round De Gea and finished into the empty net in the 91st minute.
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Atletico, who have won the cup nine times, were denied a second trophy of the campaign after last week's 2-1 victory over Fulham in the Europa League final.
"Antonio Puerta was with us tonight," Alvarez said in a television interview.
"You couldn't tell that our fans were in the minority and they helped us as if it were our own stadium," he added.
"All these people who have travelled more than 1,000 kilometres to get here on a work day. It's incredible."
On a balmy evening in the Catalan capital, the match kicked off after enthusiastic singing along to the wordless Spanish national anthem, in contrast to last year's final between Barcelona and Athletic Bilbao when it was roundly booed.
The last time Barca's Nou Camp stadium hosted a cup final was in 1970, during the Francisco Franco dictatorship when the competition was called the "Copa del Generalisimo".
"It was an emot
