Dani Alves had put the hosts ahead in the 36th minute only for the referee to controversially disallow Ivan Rakitic's headed goal when there appeared to be nothing wrong with it.
Gallingly for Sevilla, Lionel Messi doubled the lead.
However, Rakitic pulled one back and Sevilla appeared to have earned a draw with a last-minute volley from Coke, only for Alexis to break their hearts at the end of stoppage time.
While Barcelona started the game with a perfect nine points from three games, their opponents had struggled and were still searching for a first victory.
Draws against Levante and Malaga had brought Sevilla their only points this season.
Barcelona started with clear attacking intent, Neymar having a drilled shot parried by Beto in the third minute.
The Brazilian then attempted to curl a right-footed shot into the net after cutting in on the left side of the area only for Beto to go full-stretch and divert the ball out for a corner.
Cristian Tello came within inches of giving Barca the lead.
Tello's first shot was blocked but the ball came back to him and his left-footed low effort from the edge of the area was tipped round the post by Beto.
Messi came closest to putting the hosts ahead when his curled free-kick beat Beto all ends up but missed the top right-hand corner by inches.
Sevilla could only repel Barcelona for so long and inevitably the hosts took the lead in the 36th minute.
It was a superb build-up which ended with Alves running on to Adriano's floated cross to the right of the penalty area, with the defender bravely heading the ball home despite having a boot in his face.
However, Sevilla missed a golden chance to equalise when the Barcelona defence was a man down after Gerard Pique had collided with a defender in the Sevilla area.
Tello played in Messi on the right of the area but he had little time and space and Beto managed to rush and block the Argentinian's effort.
Tello then forced the keeper to parry the ball after a shot across the goal from the right on the stroke of half-time.
However, while there was just one goal in it, Sevilla posed a danger and Pique had to be alert to block Vitolo's shot when the visitors broke.
The referee then made a big call when Rakitic's flicked headed goal was ruled out midway through the half although TV replays showed there was nothing wrong with it.
Messi doubled the lead in the 75th minute with a well-crafted goal.
Cesc Fabregas' pass to Neymar at the right of the area was in turn passed first-time by the Brazilian across the box with Messi firing high into the centre of the net.
However, Sevilla pulled one back with a breakaway goal less than five minutes later.
Vitolo did well, cutting in on the left before going along the top of the area and laying the ball off to Rakitic, who fired a fierce shot across the keeper, and moments later Juan Cala sent a strong header narrowly over for Sevilla.
Adriano's curled shot was parried for a corner and the same player was vital at the other end, putting the ball out for a corner as it fizzed across goal.
However, from the resulting corner Sevilla drew level, Coke left on his own to volley home.
Sevilla almost got a third but it was Barcelona who broke their opponents' hearts with seconds left.
Messi forced his way through the defence by sheer force of will and pulled the ball back with Beto's parry pounced on by Alexis, who fired the ball home.