The tie was finely poised after last week's 1-1 draw in Galicia and Celta were well in with a shout of getting through but within 15 minutes the tie was over.
Messi then returned the favour by setting up Alba to score past the hapless Sergio Alvarez in the Celta goal before Luis Suarez made it 4-0 on the night after 30 minutes when the Uruguayan burst through the defence.
"It's luxury to have Leo in the side, he's the best in history".
Suarez joined the party even before the half time whistle and capitalising on Pione Sisto's sloppy back pass, Suarez found the net for the fifth game running.
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With the tie balanced at 1-1 after the away leg, Messi starred at the Camp Nou with a pair of goals two minutes apart in the first half.
New signing Philippe Coutinho watched from the Camp Nou stands following his club-record transfer from Liverpool earlier this week.
Daniel Wass and Hugo Malla struggled to keep up with the pace of Jordi Alba and succumbed under pressure. "We always look for each other and we were both able to get on the score sheet today".
Not long after, Andres Iniesta was also withdrawn from the action, and given there was no need to chase more goals, Barcelona were happy to hold the ball and conserve energy for much of the second period.
Sevilla beat Cadiz 2-1 with goals from Wissam Ben Yedder and Joaquin Correa to advance 4-1 on aggregate.