The four-in-a-row winners of this competition were level in the tie inside 31 minutes thanks to a Philippe Coutinho penalty and an Ivan Rakitic strike, either side of a costly penalty miss from Ever Banega.
Following a spell of sustained pressure, Barca were awarded a penalty when Quincy Promes was adjudged to have tripped Messi in the box, though the Argentine forward looked to have kicked his opponent's boot.
Coutinho picked out the bottom corner of the goal from the spot, sending the ball past Juan Soriano despite the keeper guessing the right way, to give the home side a route back into the tie.
Sevilla, without a win at Camp Nou since 2010, started to grow into the match and should have been level on the night by the half-hour mark.
Barca had Cillessen to thank for still being in the lead as the back-up keeper reacted brilliantly to push Andre Silva's delicate flick onto the post, before then keeping out Banega's penalty after Gerard Pique made contact with Mesa in the box.
Four minutes later Rakitic doubled his side's tally, getting a slight touch on Arthur's defence-splitting pass from deep to help the ball past Soriano.
A good chance came and went for Pique to complete the comeback, glancing narrowly wide from a corner, but Barca ensured that the centre-back's miss would not come back to haunt them.
Coutinho sent a stooping header away from Soriano after being picked out by Suarez, and Roberto added a fourth by latching on to Messi's through-ball and sweeping the ball into the far corner.
Despite keeping top scorer Wissam Ben Yedder on the bench until the final stages, Los Rojiblancos had a route back into the match with a little over 20 minutes to go as a poor Cillessen clearance was pounced on.
Banega laid the ball off for Arana to smash it into the Barca net first time from just inside the box, setting up what could have been a tense finale.
One goal would have been enough to send Sevilla through, but Suarez converted Jordi Alba's back-post cross and Messi rounded off another swift breakaway in the final minutes to send Barca into the semis.
BARCELONA (4-3-3): Cillessen; Roberto, Pique, Lenglet, Alba; Arthur (Alena 90'), Busquets, Rakitic (Vidal 76'); Messi, Suarez, Coutinho (Semedo 81')
SEVILLA (3-5-2): Soriano; Carrico, Kjaer, Gomez; Promes, Banega (Ben Yedder 77'), Mesa (Vazquez 77'), Amadou, Arana (Gil 84'); Silva, Sarabia