An under-strength Barca side were seeking their fifth trophy of the year but went behind after a stunning drive from the centre circle by Mikel San Jose sailed over keeper Marc-Andre Ter Stegen after 13 minutes.
Aduriz headed his first after 53 minutes and then sloppy defending allowed him to drive Bilbao's third.
Dani Alves then fouled Xabier Etxeita and Aduriz sent TerStegen the wrong way from the penalty spot to complete his treble after 68 minutes.
Unsettled forward Pedro started the game and came closest to scoring for Barca with a shot against the woodwork after 50 minutes.
The return leg is at the Nou Camp on Monday but even for Lionel Messi and company it may be asking too much to turn the tie around, though coach Luis Enrique is confident they can.
"If anyone can turn this result around then it is Barca and if not we will still try with all we've got," he told a news conference "It is not being arrogant to think that we can come out on top.
"This was not our best performance but still we were not fortunate."
Barca had the better of the opening minutes but Bilbao, who also had a number of players missing through injury, began to press.
Javier Eraso's shot from a tight angle was pushed behind
for a corner by Ter Stegen and then Bilbao took the lead.
Ter Stegen came out of his area to head away a long clearance by opposite number Gorka Iraizoz and San Jose hit the ball straight back over him.
The visitors had a lot of the ball but their passing was slow. Pedro battled hard and after the break he was unlucky when his volley hit the bar.
Barca beat Sevilla 5-4 after extra-time to lift the European Super Cup on Tuesday and Luis Enrique had made several changes with Andres Iniesta, Ivan Rakitic and Gerard Pique among the
players that began the game on the bench as the coach looked to rest them after the Sevilla match.
"There were a lot of changes but it had to be that way after playing for 120 minutes," he added.
"Football is a game of errors but we have gone from a side that concedes fewest to one that concedes too easily.
"I have made changes to the side and I accept the responsibility for the loss."
(Writing by Tim Hanlon, editing by Tony Jimenez)
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