Los Blancos – aiming to move ahead of Real Sociedad and maintain the pace with rivals Barcelona, who play later on Saturday – hit three goals inside the first 30 minutes.
Federico Valverde clipped in a fourth on the hour, his first LaLiga goal for Real Madrid, to seal a comfortable win for Zinedine Zidane’s side, who followed up their midweek 6-0 Champions League thrashing of Galatasaray.
Benzema had an early chance when he collected a chipped pass from Ferland Mendy and saw his shot glance the outside of the post, but the offside flag was up.
After 17 minutes, Benzema made no mistake as he drilled home from a tight angle when the ball broke to him following Valverde’s run.
Before Eibar could regroup, they were 2-0 down. Hazard burst into the left side of the penalty area and was tripped by Pablo De Blasis.
Ramos made no mistake from the spot, sending Dmitrovic the wrong way.
The Eibar keeper was soon in the action again, this time coming out to challenge Hazard, who went down, but the referee waved play on as the ball dropped wide.
This time Benzema stepped up and rolled the ball low into the left corner.
At the other end, Dmitrovic tipped over a bending effort from full-back Dani Carvajal.
With the rain now lashing down, Eibar, who had won their two previous LaLiga matches, rallied again just before half-time, as Japan winger Takashi Inui saw his angled drive blocked.
In first-half stoppage-time, Eibar captain Gonzalo Escalante got on the end of a floated free-kick at the far post, but could only fire an angled drive over and then Sergi Enrich’s header dropped just wide.
Eibar defender De Blasis curled an effort just off target at the start of the season half.
Los Blancos, though, never looked like allowing the hosts a lifeline, and wrapped things up on the hour when Valverde produced a fine finish from 20 yards after being picked out by Luka Modric.
During the closing stages, Real Madrid keeper Courtois passed the record of Keylor Navas for more than 517 minutes without conceding a goal.
Real Madrid substitute Vinicius Junior saw a late header saved as Los Blancos – who had lost 3-0 in Ipurua last season – coasted to a comfortable win.