Benzema then set up Madrid's second goal minutes later, laying off a perfectly weighted pass for Luka Modric to hit his first-time shot past Dmitrovic to double the advantage for Real.
After the match, Madrid boss Zinedine Zidane said: "We suffered but that's normal. We started well, scored twice, then the game changed a bit, but we were very focused, we had difficulties in the game, it's a difficult place to play, and an opponent who don't stop fighting, I think they're the team that win most balls in the opponents' half."
Real Madrid continued to dominate possession, but Eibar were dangerous as well despite neither team being able to score from decent chances as the second half played out.
Japanese midfielder Takashi Inui latched on to a disastrous pass from Real captain Sergio Ramos but failed to find the target, while his compatriot Yoshinori Muto struck the woodwork soon after.
Eibar then had loud appeals for a penalty turned down when the ball struck the arm of Ramos.
However, Dani Carvajal said he didn't see the handball and added sometimes it's hard to know what is and isn't a penalty anymore.
"Honestly, it isn't clear for us," Carvajal said. "There are some that are given and some that aren't. For example, one was given against Nacho against Alaves, the other day one wasn't given against [Athletic Bilbao's Ander] Capa.
"We don't know what the criteria are based on. I think it should be clarified, for us and against us, because we the players have doubts, we appeal all of them for our side, but it's true that it isn't clear."
Lucas Vazquez added one more right before full-time to reach the final scoreline for Real Madrid.
Asked about his plans for rotating the Real Madrid squad, Zidane added: "For now, it's like that. There will be changes, it's a long season, we're rotating a bit less for now but the important thing is that the players are right,
"I'll count on all of them. We haven't made many changes lately but it's a one-off."
The result sees Real stay level on points with leaders Atleti, who have a superior goal difference over their city rivals in the table.