The hosts began brightly against their in-form opponents but were tested on 12 minutes when Daniel Wass's low delivery to the near post was met by Iago Aspas, only for Javier Mascherano to weigh in with a good challenge.
Just one minute later, the Blaugrana were ahead thanks to Lionel Messi, who was on hand to rifle Jordi Alba's cross towards goal, the ball powering through the hands of Celta goalkeeper Sergio Alvarez.
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.
The tie was effectively over by the half-hour mark as Ernesto Valverde's charges registered a third goal on the night, Messi curling a ball behind the Celta defence and into the left inside channel for Alba to finish off.
The hosts took their foot off the accelerator thereafter but continued to dominate proceedings against a subdued Celta side that struggled to create chances and often failed to string passes together.
Barca eventually claimed a fifth on the night to compound the visitors' misery, Ousmane Dembele lifting a corner into the box for Ivan Rakitic to bury home with a free header into the top corner of the net from 10 yards out.
The result ensures that the Catalan giants, who drew 1-1 with Celta in the first leg on January 4, progress to the quarter-finals with a 6-1 aggregate win.
BARCELONA (4-3-3): Cillessen; Semedo, Pique (Vermaelen 46'), Mascherano, Alba; Rakitic, Gomes, Busquets; Iniesta (Jose 63'), Suarez, Messi (Dembele 59')
CELTA VIGO (4-4-2): Alvarez; Mallo, Gomez, Fontas, Jonny; Wass (Radoja 46'), Sanchez, Lobotka, Sisto (Mendez 57'); Aspas (Gomez 46'), Mor