Barca came into the match on the back of a defeat to Levante in the Copa del Rey, but were back to winning ways with a relatively simple victory at Camp Nou.
The sale of Munir El Haddadi to Sevilla led boss Ernesto Valverde to say that they need to sign a back-up to Suarez in the January transfer window, and the Uruguay international showed just how crucial he is to the Catalans.
He opened the scoring on 19 minutes as he finished off a superb passing move to place the ball past Eibar goalkeeper Asier Riesgo.
Messi doubled his side's lead on 53 minutes as he reached another personal milestone. The Argentine received Suarez's lay off, before taking a touch to open up some space and firing low into the corner.
And six minutes later, Sergi Roberto took a quick throw-in to play in Suarez who delicately curled the ball past Riesgo with the outside of his foot.
Eibar were 13th on goal difference coming into the match, with three teams below them level on 22 points, but dropped to 16th after conceding three here.