The title holders had drawn the first leg 1-1 last week with a weakened squad, but named a stronger line-up for the home game to win 6-1 on aggregate.
Alba did well down the left and put a low cross into the box which Suarez left for Messi to sweep home after 13 minutes.
New signing Philippe Coutinho, who officially joined on Monday, watched on from the stands as Barca ran riot.
"We always look for each other to score goals and today we have done it twice", Alba said.
The Argentina global threatened for the first time on 10 minutes when he surged forward on the counter before being crowded out, but he latched onto a loose ball and tested Sergio Alvarez with a low shot from 20 yards out. "Messi also creates goals for Alba".
Worse was to follow for Celta on 31 minutes when Pione Sisto gifted the ball to Suarez inside the box with a poor backpass and the Uruguay global smashed his shot into the top-right corner of the goal.
Suarez nearly capped the win with a spectacular chip from 40 yards, with the ball just sailing wide of the left-hand post, but Rakitic added gloss to the scoreline three minutes from time with a firm header from Dembele's corner.
The visitors failed to pitch up in the first half and they were duly punished by Barca, who were out of sight at the half-time whistle.
Sevilla beat Cadiz 2-1 with goals from Wissam Ben Yedder and Joaquin Correa to advance 4-1 on aggregate.
Celta drew 2-2 in the Camp Nou in the La Liga earlier in the season and held Real Madrid to a 2-2 draw on Sunday.
Two derbies feature in the Copa del Rey quarterfinals, with Barcelona facing Espanyol and Real Madrid taking on Leganes.
Goals by forwards Leo Baptistao and Gerard Moreno helped Espanyol to a 2-0 victory at Levante, sending the Barcelona-based team through 3-2 on aggregate.