Suarez nearly capped the win with a spectacular chip from 40 yards, with the ball just sailing wide of the left-hand post, but Rakitic added gloss to the scoreline three minutes from time with a firm header from Dembele's corner.
Messi quickly quashed any chance of Celta springing an upset.
Gerard Moreno then smacked the post as Espanyol continued to threaten on the break but he added a second before the break when Tono failed to clear by booting the ball against his goalkeeper allowing Moreno to score.
Messi then turned provider, lobbing a ideal pass behind Celta's back line to meet the run by Alba, who chipped Alvarez to score.
With Messi having re-signed until 2021, Alba said he hoped the 30-year-old could maintain his incredible form.
"They look for each other and are happy to do so, because it is not just Alba assisting Messi".
Not content with their 2-0 lead, Barca continued to push forward, with Suarez seeing a goal harshly ruled out for offside before Andres Iniesta was denied by Sergio's fingertips after a fine Messi backheel.
And his call paid off in an 18-minute first-half blitz that saw Messi net twice before Jordi Alba and Luis Suarez got in on the act.
Celta Vigo faced Barcelona at Nou Camp at the Spanish League and they managed to end the game with a 2-2 draw.
Barca join Real Madrid, Atletico Madrid and Valencia in the draw for the quarterfinal, which will take place on Friday.
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