Eventually Madrid appeared to exert a degree of control and Ronaldo - following a moment of Luka Modric inspiration - opened the scoring in the 34th minute, continuing his brilliant recent form.
With 50 minutes on the clock, Ramis headed home unchallenged from the resultant corner from Pedro Leon.
This time, Dani Carvajal whipped in a cross that Ronaldo met on the run to smash into the back of the net with a strong header.
Alves led his team to reach the competitions' finals and they lost to Real Madrid 4-1 to lose the Champions League finals for the first time. He knows that he's going to score, he's very positive. He is a different, special player.
Dmitrovic produced another wonderful save to thwart Ronaldo from 12 yards after a great cross from Bale before the Welshman glanced a header just wide of the far post as a second Madrid goal looked imminent.More news: Toys R' Us Considering Liquidation
Real´s win in La Liga moves them to within four points of second-placed Atletico Madrid, who play at home to Celta Vigo on Sunday.
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.
Ronaldo couldn't poke a 54th minute shot with the outside of his boot on target following some good build-up play, while Modric probably should have passed to an unmarked Ronaldo at the back post after breaking through the Eibar defence, and pulling his shot just wide.
Madrid coach Zinedine Zidane sent on Karim Benzema and Lucas Vazquez to look for the victor.
Valencia tightened their grip on fourth place in the standings and took a giant leap towards returning to the Champions League by winning 2-0 at Sevilla, opening up an 11-point gap with the Andalusians in fifth place.