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.
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.
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.
Messi then turned provider, lobbing a flawless pass behind Celta's back line to meet the run by Alba, who chipped Alvarez to score.
The tie petered out in the second half but French worldwide Ousmane Dembele showed why Barca shelled out £138m for him by dribbling past three defenders from the halfway line before being denied by a desperate tackle.
And Alba has explained why Messi is "such a luxury". Barcelona defeated Celta Vigo by a score of 5-0 and advanced to the quarterfinals of this tournament. It often looks like Messi is playing a different game to his contemporaries and after getting his name on the scoresheet twice he set up Barcelona's third goal.
The fifth goal arrived three minutes from the full-time whistle when an un-marked Ivan Rakitic sent a fine header into the top corner following a cross from Ousmane Dembele.
Sevilla beat Cadiz 2-1 with goals from Wissam Ben Yedder and Joaquin Correa to advance 4-1 on aggregate. Unzue made six changes to the team that drew 2-2 against Real Madrid in the weekend.
Earlier, Barca's city rivals Espanyol had overturned their first leg deficit to get past Levante 3-2 on aggregate.
In a much-changed side, Lucas Vazquez scored both of Real's goals after the 11th and 59th minutes.
Numancia ended the game with 10 men after captain Dani Calvo was red-carded in the dying moments.
Moroccan global Nabil El Zhar scored the crucial away goal for mid-table Leganes on the half-hour mark.