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.
NEWS:Sevilla dedicate win to Puerta
VIDEO:Watch the goals
It was a fifth domestic cup success for the Andalucian club, winners in 2007, and ended their season on a high after they finished fourth in La Liga and qualified for the Champions League.
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.
Sevilla coach Antonio Alvarez dedicated the cup win to the fans and defender Antonio Puerta, who died in 2007 after collapsing during a La Liga match.
"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."
PARTY MOOD
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.
In the festive atmosphere, many Sevilla fans were sporting sombreros in homage to club president Jose Maria del Nido and vast swathes of the giant arena were decorated in the red and white of Atletico.
The party mood in the crowd failed to rub off on the teams and the match boiled over with around 20 minutes left when a crunching tackle by Luis Perea on Capel prompted pushing and shoving between players and staff next to the technical areas.
Atletico coach Quique Sanchez Flores and Alvarez had a heated exchange before match officials restored calm.
Atletico pressed forward desperately as the clock ticked down, leaving holes at the back that Navas was able to exploit to kill the game off.
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".
Real Madrid beat Valencia 3-1 to win the trophy and Franco was in the stadium to present the cup to Real captain Gento.
This time it was the Prince of Asturias who handed over the trophy to an ecstatic Sevilla captain Andres Palop.
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