Tito Vilanova's men appeared to be heading for defeat when Piotr Trochowski and Alvaro Negredo fired Sevilla into a deserved 2-0 lead, but Cesc Fabregas' brace pulled Barca level before substitute David Villa secured the win in stoppage time.
Sevilla were not helped by the 72nd-minute dismissal of Gary Medel following a clash with Fabregas, with Barca's final two goals coming while the hosts were attempting to hold on with 10 men.
Barca's sixth win in as many games leaves them four points clear of Malaga at the summit, while Sevilla stay in fifth after their first defeat of the campaign.
Negredo and Ivan Rakitic had early sighters as the home side started well, but it was Barca who had the first clear-cut chance when Lionel Messi linked with Xavi before firing in a low shot that Andres Palop did well to parry.
The home side went ahead in the 26th minute, but there was more than a little good fortune in the build-up.
Medel was allowed to advance on goal before pulling the trigger with a low shot that looked to be heading wide until it cannoned off Javier Mascherano and into the path of Trochowski, who took a touch before drilling the ball low beyond Victor Valdes.
Negredo and Medel had chances to extend the lead before the break, and it was no surprise to see the pair combine for Sevilla's second after 48 minutes.
Sergio Busquets gave the ball away in midfield and Sevilla profited to the full, Medel latching onto the ball and finding Negredo, who ran on before lofting a brilliant chipped finish over the advancing Valdes.
Two minutes later Sevilla had Alberto Botia to thank for keeping Barca off the scoresheet, the defender tracking back smartly and finding himself in the right place to clear Pedro's fiercely-struck volley off the line.
But in the 53rd minute Barca were back in the game, Pedro laying the ball off for Fabregas to lash home a powerful finish from 25 yards.
Alexis Sanchez then flashed wide of the target as Barca continued to pour forward, while in the 61st minute Sevilla were again indebted to Palop for a wonderful save, this time the veteran defying all of his 38 years to turn aside a Messi free-kick that seemed destined for the top-right corner.
Sevilla found themselves up against it with 18 minutes remaining, however, when the hitherto influential Medel was sent off following a needless tussle with Fabregas.
The Barcelona man riled Medel as he attempted to retrieve the ball, but as the pair came together the Sevilla midfielder motioned his head slightly in the direction of the former Arsenal player, who made the most of what contact there was and the referee duly brandished the red card.
Sevilla responded to the dismissal by withdrawing Rakitic and Negredo for Geoffrey Kondogbia and Antonio Luna as the hosts looked to pull down the shutters in the face of the expected Barca onslaught.
Villa was guilty of over-playing when he tried to square the ball when a shot was certainly on, but in the 89th minute Barcelona did find a leveller - albeit in controversial circumstances.
Thiago Alcantara appeared to handle the ball as he intercepted a pass out of the Sevilla defence, but when the referee allowed play to continue the ball eventually broke to Messi, who slotted through for Fabregas to apply a cute finish.
Sevilla coach Michel was sent to the stands for his protestations to the referee over Thiago's role in the build-up to the goal, and his mood will have soured further still three minutes into stoppage time as Barca bagged a dramatic winner, Villa exchanging passes with Messi before flashing the ball beyond the exposed Palop.