Madrid - without Ronaldo through injury - also had to cope without suspended duo Gareth Bale and Alvaro Morata, Zinedine Zidane deprived of his three starting attackers from the draw with Las Palmas last time out.
But Benzema, who had not scored in his previous seven LaLiga games in his worst top-flight run for three years, struck twice to send the visitors on their way to a vital win in the title race.
James Rodriguez then got his first league goal for five-and-a-half months as Zidane's men opened up a three-goal lead within 29 minutes.
Marco Asensio capitalised on his first league start of 2017 with Madrid's fourth after the break, before Ruben Pena scored a consolation for Eibar.
Eibar had been chasing what would have been a club-record fourth straight home league win, but stay seventh after a convincing defeat.
3 - Karim Benzema hadn't had a hand in three goals in a single game since January 2016 vs Sporting in La Liga (2 goals, 1 assist). Claws— OptaJose (@OptaJose) March 4, 2017
Madrid took the lead after 14 minutes when Asensio - in as one of Zidane's eight changes - had exchanged passes with James and sent in a low cross from the left.
Benzema's initial shot from a tight angle was parried by Yoel, but the striker kept his cool and found the bottom corner with a left-footed strike on the rebound.
The forward netted again in the 25th minute after being left completely unmarked to convert James' in-swinging free-kick from inside the six-yard box.
Benzema then turned provider as Madrid struck again four minutes later, collecting a pass from Luka Modric before seeing his right-wing cross impressively converted by a sliding James, who directed his finish into the top corner.
Yoel got down well to keep out efforts from Benzema and Asensio as the visitors went into half-time in complete control.
Keylor Navas had still not been troubled by the time Madrid scored their fourth after an hour and Benzema was involved again with his chip sending Asensio racing towards goal.
He then squared for James, whose curling effort bounced of the post before Asensio reached the rebound and produced an impressive left-footed finish from a difficult angle.
With the result beyond doubt, Benzema was given a break for the closing stages, Mariano given a rare chance to impress.
Eibar had a goal ruled out for offside as they attempted to respond, Gonzalo Escalante flagged for offside having tapped in after Navas had kept out Sergi Enrich's header from point-blank range.
But the hosts got a goal that did count with 18 minutes remaining, a cross from ex-Madrid winger Pedro Leon - who the visitors claimed had fouled Nacho in the build-up - turned in by Pena at the far post.
Any hopes Eibar had of a positive result had long been extinguished, though, as Madrid put the pressure back on Barca and turned their attention to Tuesday's crucial Champions League last-16 second-round match with Napoli, which they lead 3-1.
1 - Eibar have scored in La Liga against Real Madrid at Ipurua for the first time ever. News— OptaJose (@OptaJose) March 4, 2017