Alba had provided the assists for Messi's strikes in the 13th and 15th minutes, and the Barcelona star returned the favor by setting up the left back to score in the 28th.
A 1-1 draw from the first match in Galicia had left the tie looking open, but a devastating first-half performance from the holders secured their 6-1 aggregate victory.
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Barcelona: Jasper Cillessen 5 (out of 10), Nelson Semedo 6, Gerard Pique 6, Javier Mascherano 6, Jordi Alba 7, Ivan Rakitic 6, Sergi Busquets 7, Andres Iniesta 7, Andre Gomes 6, Lionel Messi 8, Luis Suarez 7.
Given Celta have performed well against the better sides in La Liga this season, there was no sense that this tie was over after the first leg.
Messi was on hand to double Barca's lead on the quarter-hour mark after playing Alba into space, who in turn charged down the wing and played the ball back into the path of the Argentine, who duly drilled home.
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"It's a luxury to have Leo in the side", Alba told Gol Television.
And Messi then turned provider for the Spanish defender with a sumptuous pass to get him on the scoresheet as club record signing Philippe Coutinho watched from the stands in the Camp Nou.
It was a vintage Messi display, too, and despite netting a couple of goals early on, the finest moment of the first half was his involvement in the third.
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Celta looked in disarray and they gifted Barca their fourth three minutes later, with Pione Sisto passing straight to Suarez 12 yards out, allowing the striker to fire home.