Karim Benzema was the focal point in Real's attack without the suspended Bale and injured Ronaldo and fired the visitors ahead in the 14th minute, following up his own shot on the rebound, before stretching their advantage in the 25th.
James Rodriguez provided the cross for Benzema's second goal and the Frenchman returned the favour for his Colombian team mate to net his first league goal since September in the 29th minute.
Marco Asensio hit the fourth on the hour with a rebound after James had struck the post.
Ruben Pena bundled in a consolation goal for Eibar in the 72nd minute but Real still recorded their biggest win since beating Real Sociedad 3-0 in January.
The victory will lift Real's confidence after dropping five points in three games. They top the standings on 59 points although Barcelona, on 57, can return to the summit when they host Celta Vigo later on Saturday.
Zinedine Zidane made eight changes to the team that drew 3-3 with Las Palmas on Wednesday. He was forced to do without Ronaldo, who had scored twice against Las Palmas to salvage a point, because of ankle injury.
Midfield anchor Casemiro returned after being rested against Las Palmas to the team's detriment. He brought stability in midfield while the returning Luka Modric added flair and style, and was even applauded by the Eibar fans.
"It was a very good reaction, we scored three times in the space of half an hour. We knew it was a difficult ground to go to but we scored early and gave a serious display," Zidane told reporters.
"All sorts of things happen during a season and a team can't always play well, but we did very well today and played with a lot of intensity. We made a flying start and that was very important. We know that there's still a long way to go."
(Reporting by Richard Martin, editing by Tom Hayward and Neil Robinson)
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