opener was cancelled out by Ivan Ramis's header, but the Portuguese star netted six minutes from time to seal all three points for Zinedine Zidane's men.
Madrid - who lost Raphael Varane to a head injury - gained a foothold, though, with Gareth Bale forcing the approaching Marko Dmitrovic into action in the 32nd minute.
"This is a very hard ground to come to and they know how to play here".
After almost half an hour, Real's coach Zinedine Zidane was forced to make the first change, bringing in Nacho for Raphael Varane, who left the field after a skull-to-skull collision with Kike Garcia.
After the intermission, and with a goal in their pocket, Real Madrid tried to slow the game's pace, but Eibar did not allow them to.
Real Madrid have closed the gap on second-placed Atletico Madrid with a hard-fought 2-1 win in Saturday's La Liga clash at Eibar.
After securing a 5-2 aggregate victory over Paris Saint-Germain in the Champions League last-16 on Tuesday, Real swapped the Parc des Princes for the smallest ground in La Liga, with a capacity of just over 7 000.
Eibar appeared to target Madrid's full-backs and their crosses from both flanks continued to trouble, even if they often just missed the intended target.
Lucas Vazquez and Karim Benzema replaced Isco and Toni Kroos as Zidane looked to raise the tempo and find a second goal.
And Zidane was impressed with Real's character against an Eibar side who were on top for long periods of the match.