Benzema has been one of the few bright sparks in an otherwise dreary campaign for the European champions and it was his second-half treble that proved the difference at the Santiago Bernabeu.
Victory moves Madrid four points behind Atletico in second, while Bilbao stay seventh, with their chances of sealing a spot in the Europa League dented. However Zinedine Zidane's side stay 13 points behind La Liga leaders Barcelona, who beat Real Sociedad 2-1 on Saturday.
Benzema's advance in the absence of Cristiano Ronaldo and surge since the reappointment of Zidane comes in stark contrast to Gareth Bale, who was whistled when introduced as a substitute here and frequently during his 20 minutes on the pitch.
For all the impatience of the home crowd, it was Bale who won the corner for Benzema's second and then teed up his team-mate's third in injury-time.
But his sixth season at Real Madrid will surely be his last, with Zidane saying little in press conference either to express support or suggest he wants the Welshman to stay beyond the summer.
Bale and Benzema with the gargantuan task of filling the void left by Ronaldo's departure for Juventus last summer and while Bale seems to have shrivelled under the weight of expectation, Benzema has thrived.
The 31-year-old has now claimed all of Madrid's last eight goals in La Liga, a run that includes five matches, while in his last seven games, he has scored 10.
Benzema saves limp Real
Benzema said at the start of the season he wanted 35 goals in all competitions and, despite his opportunities being reduced by his team's early exit in the Champions League, he is now only five short of his target with the same number of games left to play.
Zidane has already said he expects Benzema to be a Madrid player again at the start of next season and on this evidence it is easy to see why.
The striker's clinical performance aside, the reigning European champions lacked spark in the final third and, the emphatic scoreline flattered them.
They were particularly underwhelming in the first half and Bilbao's Inaki Williams might have scored early in the second had the 22-year-old Jesus Vallejo not made an excellent last-ditch intervention.
It was from the counter-attack that Madrid scored as Toni Kroos was allowed to stroll forward unattended and feed Marco Asensio out wide. Asensio crossed for Benzema to head home.
The game was reduced to almost a walking pace around the hour, with the Madrid fans roused only to jeer the introduction of Bale, who came on with Isco, with 20 minutes left.
But it was his run in behind that won the corner for Benzema's second, another header, this time at the back post, before a comical mix-up in the Bilbao defence gifted him his hat-trick.
Iago Herrerin rushed out to intercept a speculative ball over the top but made a mess of his headed clearance, the ball falling to Bale to tee up Benzema, 30 yards out but with the goal open.