Zinedine Zidane's men were out to produce a response following Wednesday's defeat at Valencia, and although their collective performance was underwhelming, Benzema proved decisive to give Los Blancos' coach a win in his 100th LaLiga game at the helm.
Feisty Eibar deservedly held a half-time ahead, with Barcelona-owned Marc Cardona rounding off a lovely move with a clever finish.
The hosts' pressure told after the break, however, with Benzema - who previously had two goals disallowed - netting with a towering header to restore parity before repeating the feat late on to secure all three points.
Madrid looked livelier during the early exchanges and did find the net in the ninth minute, but Benzema's celebrations were abruptly halted as Gareth Bale – who provided the pass – had strayed offside.
Yet the hosts fell behind shortly before half-time, as Gonzalo Escalante carved open the Madrid defence with a brilliant pass to pick out the run of Cardona, who flicked over the helpless Keylor Navas.
Madrid were again denied by the offside flag early in the second half - VAR confirming Benzema made his run too early before making the most of a defensive mix-up to put into an empty net.
It was third time lucky for the Frenchman, though, when he met Marco Asensio's cross with a header which bounced up off the ground and into the roof of the net.
Benzema completed the turnaround nine minutes from time, rising to meet Toni Kroos' right-wing delivery and picking out the bottom-left corner.
He should have completed his hat-trick towards the end but missed two sitters, while a post denied him another goal in added time, though it did not impact the result.
Benzema's persistence pays off
Although he had two goals ruled out, Benzema still deserves credit for his leading of the line and almost singlehandedly preventing an embarrassing defeat. He kept the Eibar centre-backs occupied throughout and his two well-taken headers were just rewards for his efforts.
Bale fails to convince
After Zidane's somewhat cryptic comments about Bale's future at the club, this was an opportunity to really earn his place for the rest of the season. However, he did not have a huge impact, with his straying offside for Benzema's first disallowed goal particularly frustrating, given it was not necessary for him to go so early.
