Lionel Messi had given the European champions -- who trailed 4-0 from the first leg -- a glimmer of hope with the opener two minutes before half-time on Monday.
However, any realistic hope Barca had of pulling off a monumental comeback disappeared when Gerard Pique was shown a straight red card for dissent 10 minutes into the second-half.
And Aritz Aduriz exploited the space left by the Spanish international defender's dismissal to follow up his hat-trick in the first leg with the crucial away goal midway through the second-half.
The hosts started on the front foot as Luis Suarez saw a shot deflected just over before Pique's effort came back off the crossbar.
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Yet, Athletic's mass ranks of defence were comfortably keeping Enrique's men at bay and the visitors even had a great chance to put the tie to bed when Javier Eraso skipped past Javier Mascherano before firing into the side-netting rather than squaring for Aduriz to apply a simple finish.
Claudio Bravo kept Barcelona's hopes alive early in the second period with saves from Markel Susaeta and Aduriz, but the Chilean's good work was undone when Pique was sent-off for something he said to the referee's assistant after he failed to flag Aduriz offside.
To their credit Barca didn't give up as Pedro Rodriguez, Ivan Rakitic and Suarez all fired off target with fine chances to reduce the deficit to two goals.
However, Athletic struck the final blow when Aduriz was given acres of space to latch onto Eraso's flick-on and he slotted into an empty net at the second attempt after Bravo had saved his first effort.
Defeat ends Barcelona's hopes of matching the club record set by Pep Guardiola's side in 2009 of winning six trophies in a calendar year.
They won't have long to wait for a chance at revenge, though, as the two sides square off once more in their opening game of the La Liga season on Sunday in Bilbao.