Bilbao won the first leg 4-0 but Barcelona threatened a comeback with Lionel Messi's close-range finish.
Barca defender Gerard Pique was sent-off after the break and Aritz Aduriz struck late on to seal it for Bilbao.
The visitors had Kike Sola sent-off late on, but the celebrations for Bilbao had already begun.
It was ultimately a disappointing display for Barcelona who, despite plenty of possession, failed to threaten often enough. After Barcelona's humiliating defeat in the first leg, Athletic Bilbao knew Luis Enrique's side would be determined to make amends.
The reigning European champions lined up with Messi, Luis Suarez and Pedro in attack but the visitors had done their homework and the trio were well marshalled throughout.
Whenever Messi had the ball, at least two players crowded round the Argentina international and his inability to get into the game left Suarez isolated.
Pedro had Barcelona's better opportunities in the opening stages but was unable to make them count.
The forward was full of running but his finishing let him down when it mattered, most notably when he missed the ball completely when a corner routine was worked to him on the edge of the area in the first half.
Pedro failed to find the target with another chance after the break and was eventually replaced in the 68th minute.
Barcelona's attacking intent often left them exposed at the back, and any hopes of getting the goals they needed ended when Pique was needlessly sent-off midway through the second half. The defender angrily remonstrated with the linesman and the referee produced a straight red.
After a brief spell of Barca dominance, Jeremy Mathieu gifted possession to Aduriz, leaving the forward one-on-one with Bravo and, after his initial effort was saved, Aduriz side-footed home the rebound to put the game beyond Barcelona.
Culled from BBC Sports