Lionel Messi scored a first-half penalty and took his incredible tally for the season to 43 goals with a second-half header as Barca stretched their lead at the top of La Liga to four points.
Second-placed Real will be expected to cut that gap to a point by beating Levante on Sunday, but momentum is with the Catalan giants in the battle for the title ahead of El Clasico at Camp Nou next Sunday.
Before then Barca host City in the second leg of a UEFA Champions League round of 16 tie they lead 2-1 after the first leg.
Luis Enrique said: "This will be a very nice week with two games in front of our fans, but it is not the most important.
"We're going into the Clasico as leaders and in perfect form. But first we have to focus on City because qualification is at stake. Then we'll see what happens at the weekend."
Barca were not at their fluent best in their first trip to Estadio Municipal de Ipurua, but Luis Enrique gave credit to an Eibar side who have lost eight games in a row and backed them to avoid relegation in their first season in the top flight.
He added: "You must try to be dominant and effective in every match. Our opponent didn't make it at all easy for us, and it's essential to be effective in cases like this.
"Eibar didn't look like a team that has just lost seven in a row. They played good football and they fully deserved their great start to the season. I don't think they'll have any trouble avoiding relegation.
"Perhaps we rushed things at times but the way the game went, I think we played well and took it as seriously as we should have."
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