His double means Ronaldo has now scored in seven consecutive games and has a remarkable 13 goals overall in that period.
Real move four points behind second-placed Atletico Madrid, who will restore that gap to seven with a win over Celta Vigo today. Eibar stay eighth.
After securing a 5-2 aggregate victory over Paris Saint-Germain on Tuesday, Real swapped the Parc des Princes for the Ipurua Municipal Stadium, the smallest in La Liga with a capacity of just over 7,000.
They were below their best and rode their luck late on when Eibar should have converted a pair of excellent chances.
But Ronaldo proved the difference again, the 33-year-old taking his tally for the season to 35 goals in 37 appearances.
Isco and Casemiro formed the top and tail of a midfield diamond.
Raphael Varane had to be replaced by Nacho after taking a bang to the head and, against the run of play, Real edged ahead in the 33rd minute.
A sloppy pass out from the back by Anaitz Arbilla conceded possession to Luka Modric who quickly picked out Ronaldo with a sublime bending pass. Ronaldo chested the ball down and slammed it in at the near post.
In front, Real improved but five minutes into the second half and Eibar were level. Sergio Ramos lost his man at a corner and Ramis nodded in from six yards.
Ronaldo should have restored the advantage but failed to convert an excellent Bale cross and from the resulting corner, Bale flashed a glancing header just wide of the post.
Isco and Kroos made way for Benzema and Lucas Vazquez with 19 minutes left but Eibar had two golden chances to score a second.
First, the unmarked Inui headed over from seven yards out and then Kike hit the side-netting with only Keylor Navas to beat. They would soon prove costly misses.
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