Messi grabbed the opening two goals in the 13th and 15th minutes, both finished with a sublime first touch with his left foot, both created by Jordi Alba.
Alba has now set up seven of Messi's 22 goals this season, while Messi has set up both of Alba's strikes.
Barcelona put on a real show at Camp Nou on Thursday as they ran the riots against Celta Vigo last night.
With new signing Philippe Coutinho watching on from the stands, Lionel Messi led the destruction of the Galician side.
Ivan Rakitic added the fifth goal in the 87th minute as they completed the demolition.
Alba's cross from the left was missed by Suarez, allowing Messi to send a fine half-volley into the bottom-right corner from the edge of the area, despite Sergio's best efforts.
The Argentina worldwide 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.
Not long after, Andres Iniesta was also withdrawn from the action, and given there was no need to chase more goals, Barcelona were happy to hold the ball and conserve energy for much of the second period. Suarez was on to it quickly and blasted in the fourth goal. Celta had drawn 2-2 at Barcelona in La Liga last month and had gone into Thursday's match after holding Madrid to a 2-2 draw in the league last weekend.
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. "We are all suffering from the derby loss", Correa said.
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.
Copa del Rey holders Barcelona will take on city rivals Espanyol in the quarter-finals of this season's competition, while Real Madrid also face a tie against nearby opposition in the form of Leganes.