Though Barca showed fight to twice get back on terms, Iker Muniain's late penalty was one bridge too far to recover from.
It was quite the start at San Mames, with Muniain lighting the blue touchpaper by putting his side ahead with just two minutes on the clock.
It was a fine curling finish from the 29-year-old following good work down the right by the industrious Nico Williams.
But anything Muniain could do, so too could Ferran Torres.
Barcelona had barely been in the game but receiving the ball from Sergio Busquets on 20 minutes, Ferran turned and lashed into the far corner giving Julen Agirrezabala no chance.
With Nico Williams continuing to cause chaos down the right the hosts could quite conceivably have gone into the break ahead, but they had to wait until the last five minutes of normal time to get their just desserts.
It looked like it was over for the Blaugrana but somehow, from somewhere, they dug out a scarcely-deserved equaliser through Pedri.
It took the game to extra-time, whereupon the hosts forced their spot-kick after a handball from Jordi Alba, and Muniain showed nerves of steel to slot past Ter Stegen.
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