Madrid suffered a 3-0 reverse on its previous visit to Ipurua 12 months ago but was a comfortable winner in this clash thanks to the impressive form of Benzema, who made it nine league goals for the season.
The 31-year-old gave the visitors a 17th-minute lead and doubled his tally from the penalty spot before the half-hour mark after Sergio Ramos had also converted from 12 yards.
Federico Valverde added a fourth for Zinedine Zidane's men, who climbed above Barcelona and surprise package Real Sociedad into top spot, although Bracelona returned soon after thanks to a 4-1 cruise against Celta Vigo.
Madrid had won only two of 10 away league games since Zidane's return as coach but did not take long to assert its authority in the Basque Country, Valverde teeing up Benzema to convert the opener from close range.
Ramos tucked away the first of his side's penalties three minutes later after Pablo de Blasis stuck out a leg and tripped Eden Hazard inside the box.
Madrid put the game out of Eibar's reach with 29 minutes played, Benzema matching Ramos in sending Marko Dmitrovic the wrong way from the spot after Cote's challenge on Lucas Vazquez.
After a slow start to the second half, Madrid soon clicked into gear and had a fourth goal through Valverde's side-footed finish into the bottom-right corner.
Eibar struggled to muster much of a fight in the remainder of the match as Zidane's men claimed a fifth clean sheet in a row in all competitions.