Benzema reach the 21 league goals this season, including 10 goals in his last seven appearances for the club.
The victory left Real Madrid, who had drawn with Leganes in their last outing, in third place on 64 points. They are 13 points behind leaders Barcelona and four behind city rivals Atletico.
Zinedine Zidane once again left Gareth Bale on the bench, with Lucas Vazquez and Marco Asensio supporting Benzema in attack.
They initially cut frustrated figures throughout an opening period of few chances, with the Bernabeu crowd voicing their frustration.
However, the fans were soon serenading Benzema two minutes after the break when he opened the scoring.
Benzema doubled the score 15 minutes from the end, heading in Luka Modric’s corner after a calamitous misjudgement from Bilbao goalkeeper Iago Herrerin.
The Frenchman completed his hat-trick by firing into an empty net during the injury time.