It finished 5-0 at the final whistle as Celta could not recover from an early shock and lost the tie 6-1 on aggregate.
In the 28th minute, Messi sent a curling pass from deep towards Alba who duly obliged and slid to send the ball at the back of the net.
With Philippe Coutinho to come into this side, there are no signs of this remarkable campaign losing momentum for Barca.More news: PM congratulates ISRO team for the successful launch of its 100th satellite
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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.
Barcelona coach Ernesto Valverde took no chances after the 1-1 draw in the first leg, starting Messi in his first appearance in the competition this season after resting the Argentina forward for the first three cup games.
Valverde's chosen squad for the game includes Jasper Cillessen, Marc-Andre ter Stegen, Nelson Semedo, Gerard Pique, Javier Mascherano, Thomas Vermaelen, Jordi Alba, Lucas Digne, Sergi Roberto, Sergio Busquets, Ivan Rakitic, Iniesta, Paulinho, Andre Gomes, Carles Alena, Messi, Suarez, Ousmane Dembele and Jose Arnaiz.
"We always look for each other to score goals and today we have done it twice". The first Barcelona goal was a result of poor goalkeeping by Alvarez while the second was a fine team goal.
Sevilla also progressed with a 2-1 win at home to second-tier Cadiz for a 4-1 aggregate.