Aduriz opened the scoring from the penalty spot in the 13th minute and then doubled the visitors' lead in the 27th minute with a fantastic strike.
Although Youssef El Arabi pulled one back eight minutes after the interval, Bilbao held on to halt a run of three successive defeats in all competitions.
It helped Los Leones move away from the Primera Division drop zone and boss Marcelo Bielsa will be hoping the win can kickstart their season.
After finishing as runners-up in both the Copa del Rey and Europa League last season, Bilbao were touted as outside hopes for a Champions League place.
Yet they have suffered five defeats in the league already and had picked up just one point from four matches on the road, yet they were gifted the lead on Sunday.
With the ball running away from him, Aritz Aduriz was brought down by Pape Malickou Diakhate in the area and referee Cesar Muniz Fernandez awarding the spot-kick.
Aduriz picked himself up and slotted his penalty into the bottom right corner in the 13th minute.
Bilbao keeper Gorka Iraizoz saved a weak free-kick from Gonzalez Dani Benitez before Bilbao further strengthened their position after 27 minutes.
Aduriz picked the ball up just outside the area before drilling low past Antonio Tono.
Granada responded well, but lacked a penetrating force up front, with both Youssef El Arabi and Gomez Ibai shooting tamely at Iraizoz.
They continued their attacking approach after the break and Yacine Brahimi struck the post after being picked out by the influential El Arabi, who got the hosts back into the match in the 53rd minute.
The Morocco international collected the ball 25 yards from goal before firing past Iraizoz.
Yet, it was Bilbao who went closest to scoring thereafter, with Aduriz denied a hat-trick when his header from Gomez Ibai's corner came back off the post, with Mikel San Jose firing the rebound wide.
Tono then produced marvellous close range saves from Oscar De Marcos and Gabriel Torje, while Iker Muniain's effort was tipped over.
And the keeper was nearly rewarded for his efforts when El Arabi played through Antonio Floro Flores, but the substitute could not beat Iraizoz at his near post.