Zinedine Zidane's men were left seven points behind bitter rival Atletico Madrid in second after its win over Eibar, but Los Blancos just about did enough thanks to Benzema at Santiago Bernabeu in a match that will not live long in the memory.
Madrid was especially poor and lacking in ideas against an Athletic side up for a scuffle during a woeful first half that did little to capture the imagination.
But the breakthrough did arrive early in the second half courtesy of the in-form Benzema, and he added two well-taken finishes late on to wrap up a fairly straightforward win over UEFA Champions League-chasing Athletic.
A frustratingly scrappy first half passed with precious little excitement, as Athletic's attackers failed to link up particularly effectively and Madrid lacked cutting edge.
The most promising move at either end resulted in the only shot on target of the first 45 minutes just before half-time, as Raul Garcia's header from a corner was easy plucked from the air by Navas.
Madrid quickly stepped things up after the break, however, as Benzema rose to meet Marco Asensio's cross with a header that was too powerful for Iago Herrerin to keep out.
Benzema was taking the game to Athletic almost single handedly and went close again just before the hour, receiving Lucas Vazquez's cut-back and putting a first-time effort agonisingly wide of the bottom-left corner.
The Frenchman found the target 13 minutes from time, however, nodding in a Luka Modric corner to make the most of Herrerin's poor positioning.
Herrerin was culpable again in stoppage-time, coming way out of his goal and botching a clearance, allowing substitute Gareth Bale to tee up Benzema to seal his hat-trick with a 30-yard finish into the open goal.
Although it is hardly a huge achievement, Madrid can certainly still steal second place from Atletico in its final five games, with Diego Simeone's men just four points ahead.