Ronaldo put the European champions ahead in the 34th minute following a lovely pass from Luka Modric.
Eibar battled back after the break and levelled up on 50 minutes when Ivan Ramis headed home a corner.
But a defensive error allowed Dani Carvajal to cross and Ronaldo headed in.
Ronaldo now has 17 goals in his last 10 games in all competitions and his 18 league goal tally has him closing in on Lionel Messi who has 24.
Eibar's goalkeeper Marko Dmitrovic first prevented Bale from getting Real on the scoreboard, while defender Ander Capa anticipated Ronaldo's move to eliminate the danger.
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. What he does, is always very positive for us; he knows he is going to score. He is a different, special player.
"He's an important player and there are other players in the team that are also important", Zidane told a news conference ahead of his side's Liga game away to Eibar on Saturday.
Zinedine Zidane's side are already out of the title race in La Liga, with Barcelona opening a 15-point lead against the defending league champions at the top of the table.
They were below their best and rode their luck late on when Eibar should have converted a pair of excellent chances.
Isco and Kroos made way for Karim Benzema and Lucas Vazquez with 19 minutes left but Eibar had two golden chances to score a second.
Now, here are four reasons why Zidane's men could be celebrating a third straight Champions League title at the end of the season.
Real Madrid's form prior to the last 16 clash against the French giants raised many questions as to whether they could withstand the attacking strength of Unai Emery's side.