Goals from Daniel Alves and Lionel Messi were cancelled out by Ivan Rakitic and Coke's effort, before the Chilean's winner in stoppage-time
Dani Alves and Lionel Messi helped the Blaugrana into a cushioned lead but, with 10 minutes to go, Ivan Rakitic pulled a goal back for the visitors before Coke levelled with full-time imminent.
Both sides pushed for a last-ditch winner, with Unai Emery's men keen to crown a dream comeback, but it was the hosts who found it, with Messi's mazy run setting up Alexis to seal the three points.
Barca made just two changes from the team that picked up three points away to Valencia prior to the international matches courtesy of a Messi hat-trick, with Cesc Fabregas and Pedro dropped in favour of Xavi and Cristian Tello.
Meanwhile, Sevilla, who went into the match without a league win this season, made four changes from their 2-2 home draw against Malaga as Javi Varas, Diego Perotti, Carlos Bacca and Bryan Rabello were replaced by Beto, Vitolo, Jairo Samperio and debutant Sebastian Cristoforo.
Emery’s team were on the back foot from the start and Neymar almost gave Barcelona the lead in the third minute as he shot across goal, but Beto tipped his effort around the post - a recurring theme in the first 15 minutes.
Sevilla eventually grew as an attacking force, but at the other end of the pitch Neymar continued to wreak havoc. Visiting right-back Coke had no answer to the Brazilian’s close ball control and he set Tello up for two chances in quick succession, but Beto saved on both occasions.
For all their good defensive work, though, Sevilla could not keep Barcelona at bay until half-time as Busquets crossed from deep and Alves arrived at the back post with a late burst to head home against his old club in the 36th minute.
After growing into the match again, Sevilla thought they had levelled in the 64th minute as Cala headed Rakitic’s corner past Victor Valdes, but the centre-back was cruelly adjudged to have fouled Gerard Pique in the build-up.
Barcelona escaped that scare and Messi soon doubled their lead as he burst in front of Fernando Navarro and knocked the ball in from close range after Neymar's low cross.
Sevilla refused to lay down and Rakitic pulled one back with 10 minutes left as he fired low past Valdes after a magnificent run from Vitolo.
And just when Barcelona looked like they would hold on for the three points, Coke volleyed home Rakitic’s corner in stoppage-time as they seemed to have earned a point.
The drama, though, was far from over and Messi had other ideas as he raced to the byline and squared across the box, where Beto could only divert the ball into the path of a grateful Alexis to win it for Barca.