Suarez pounced on an ill-advised, deflected back pass, finishing easily in the 31st minute to make it 4-0 on the night and 5-1 on aggregate.
After an even first leg finished 1-1 at Balaidos, Barca tore into the Galicians in a stunning opening 30 minutes here, with Messi grabbing a brace, Jordi Alba netting a third and Luis Suarez scoring a fourth to put the hosts 4-0 up at the interval.
Perhaps unsurprisingly, Messi made way before the hour mark for Ousmane Dembele.
In the 28th minute, Messi sent a curling pass from deep towards Alba who duly obliged and slid to send the ball at the back of the net.
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 first Barcelona goal was a result of poor goalkeeping by Alvarez while the second was a fine team goal.More news: Big Boob: Lingerie Company that Supplies UK Monarch Stripped of Royal Warrant
With Philippe Coutinho to come into this side, there are no signs of this remarkable campaign losing momentum for Barca.
Barca took their foot of the gas in the second half with Messi and Iniesta both being replaced with one eye on Sunday's trip to the Anoeta to face bogey team Real Sociedad and created few chances although Suarez did hit the post. 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.
So far, Real Madrid, Atletico Madrid, Valencia, Leganes and Deportivo Alaves have punched their quarterfinal tickets.
Ben Yedder scored from a rebound after a shot by Correa to give Sevilla the lead in the 32nd. "We are sad, but we will get through this together".