كتب : Hatem Maher
Ronaldinho's goal early in the first-half was cancelled out by Alexander Kerzhakov seven minutes before the interval.
Brazilian full-back Alves secured the precious win on the stroke of the hour mark.
The victory placed Sevilla on top of the league table with 50 points, one ahead of second-placed Barcelona.
Barcelona boss Frank Rijkaard opted to leave Samuel Eto'o on the bench, fielding Ronaldinho as a lone striker supported by Ludovic Giuly and Lionel Messi on the flanks.
Unmarked Ronaldinho put the visitors in front in the 14th minute when he headed Gianluca Zambrotta's pinpoint cross past Sevilla keeper Andres Palop.
Television replays showed that Italian right-back Zambrotta was in a clear offside position when he received the long ball in the right-flank.
Sevilla's hopes of grabbing the equalizer suffered a blow after defender Aitor Ocio was sent off for tackling down Ronaldinho inside the area whilst he was on one-on-one situation.
The Brazilian playmaker failed to extend the lead as his penalty kick was blocked by Palop's legs.
Barcelona paid for the missed penalty in the 38th minute when Kerzhakov controlled Alves's short pass inside the area before brilliantly turning and hitting a low drive into the bottom corner of the net.
Alves was a constant threat throughout the game with his pace and dazzling skills.
He fired Sevilla ahead in the 60th minute when his free-kick deflected off the wall and went past Barcelona keeper Victor Valdes.
French winger Giuly controversially received his marching orders two minutes later for tugging a Sevilla midfielder.
Rijkaard introduced Eto'o instead of Rafael Marquez in a bid to claim a draw but the Cameroonian striker rarely had an impact following Zambrotta's dismissal for a second bookable offense after making a reckless challenge.
Sevilla could have added the third in stoppage time as Malian striker Frederic Kanoute's effort was collected by Valdes and Alves's fierce shot from close-range rattled against the crossbar.