The result ensures that the Catalan giants, who drew 1-1 with Celta in the first leg on January 4, progress to the quarter-finals with a 6-1 aggregate win.
Messi quickly quashed any chance of Celta springing an upset.
Barca continued to dominate and bagged a third goal in the 28th minute when Messi turned provider to pick out Alba who converted at the far post. "Messi also creates goals for Alba". "They look for each other and are happy to do so, because it is not just Alba assisting Messi".
Suarez stretched the lead in the 31st by capitalising on Pione Sisto's sloppy back pass to score for the fifth game running whereas Ivan Rakitic completed the rout with a header in the 87th against ex-Barca assistant coach Juan Carlos Unzue's team.
Real Madrid defeated Numancia to reach the quarters and will face Leganes in the next stage of the competition, while rivals Atletico Madrid are set to take on Sevilla in a two-legged duel between two top-flight titans.
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Earlier in the day, Espanyol overturned a 1-2 first leg deficit against Levante to qualify 3-2 on aggregate thanks to a 2-0 win away at Ciudad de Valencia Stadium.
Alvaro Garcia got second-division Cadiz's consolation goal in the 86th, leading to jeers from Sevilla's home fans for the team's shaky finish to the match.
Even when Barcelona were comfortably poised at 2-0, their pace did not drop down.