The Brazil forward converted a 74th-minute penalty after Sevilla left-back Benoit Tremoulinas handled inside the penalty area but Barca could not level.
Striker Michael Krohn-Dehli put Sevilla ahead in the 52nd minute and he then set up midfielder Vicente Iborra to score for the hosts six minutes later.
Sevilla's victory was their first at home against Barca in seven years.
Barcelona have now lost consecutive away games in La Liga following their 4-1 defeat to Celta Vigo on 23 September.
Luis Enrique's side have 15 points from seven games, a point behind leaders Villarreal who are away to Levante on Sunday.
Also on Sunday fourth-placed Real Madrid, on 14 points, take on city rivals Atletico Madrid who are one place and two points below them.
Elsewhere on Saturday, Granada held Deportivo La Coruna to a 1-1 draw.
The La Liga season may only be seven games old but there are signs that Barcelona's defence of their title might not be as straightforward as they hoped.
Barca are still serving a 14-month transfer ban imposed by Fifa after being found guilty of breaching rules on the transfer of players aged under 18.
They added defender Aleix Vidal and midfielder Arda Turan to their squad in the summer, but neither player is eligible to play until January.
Coach Enrique will hope they can inject fresh impetus into a side that perhaps feels a little stale without new blood.
Barcelona are without Lionel Messi until the end of next month as he recovers from the knee ligament injury picked up in last weekend's 2-1 win over Las Palmas.
The Catalan club made hard work of their opponents on that occasion and had to come from behind to beat Bayer Leverkusen by the same scoreline in the Champions League during the week.
Against Sevilla the defending La Liga champions were a little unfortunate as they struck the post five times at the Sanchez Pizjuan Stadium.
Nevertheless Barca missed the Messi's brilliance and clinical finishing despite the presence of Uruguay forward Luis Suarez and Neymar.
Match ends, Sevilla 2, Barcelona 1.
Second Half ends, Sevilla 2, Barcelona 1.
Attempt missed. Neymar (Barcelona) right footed shot from outside the box is too high. Assisted by Ivan Rakitic.
Neymar (Barcelona) wins a free kick in the attacking half.
Foul by Coke (Sevilla).
Foul by Ivan Rakitic (Barcelona).
Fernando Llorente (Sevilla) wins a free kick in the defensive half.
Attempt blocked. Ivan Rakitic (Barcelona) right footed shot from outside the box is blocked. Assisted by Sandro Ramírez.
Foul by Sandro Ramírez (Barcelona).
Michael Krohn-Dehli (Sevilla) wins a free kick in the defensive half.
Attempt blocked. Ivan Rakitic (Barcelona) left footed shot from the centre of the box is blocked.
Corner,  Barcelona. Conceded by Coke.
Substitution, Sevilla. Mariano replaces Benoît Trémoulinas.
Offside, Sevilla. Michael Krohn-Dehli tries a through ball, but Reyes is caught offside.
Attempt saved. Luis Suárez (Barcelona) right footed shot from outside the box is saved in the bottom right corner. Assisted by Jordi Alba.
Attempt blocked. Sergi Roberto (Barcelona) right footed shot from the centre of the box is blocked. Assisted by Ivan Rakitic.
Attempt blocked. Sandro Ramírez (Barcelona) right footed shot from the centre of the box is blocked.
Attempt saved. Luis Suárez (Barcelona) right footed shot from the left side of the box is saved in the centre of the goal.
Corner,  Barcelona. Conceded by Reyes.
Neymar (Barcelona) wins a free kick in the defensive half.
Foul by Vitolo (Sevilla).
Delay over. They are ready to continue.
Delay in match Benoît Trémoulinas (Sevilla) because of an injury.
Attempt saved. Sandro Ramírez (Barcelona) right footed shot from very close range is saved in the bottom left corner. Assisted by Luis Suárez.
Foul by Neymar (Barcelona).
Coke (Sevilla) wins a free kick in the defensive half.
Sergio Busquets (Barcelona) wins a free kick in the defensive half.
Foul by Fernando Llorente (Sevilla).
Fernando Llorente (Sevilla) is shown the yellow card.
Neymar (Barcelona) wins a free kick on the left wing.
Foul by Michael Krohn-Dehli (Sevilla).
Sergi Roberto (Barcelona) wins a free kick in the defensive half.
Foul by Coke (Sevilla).
Goal!  Sevilla 2, Barcelona 1. Neymar (Barcelona) converts the penalty with a right footed shot to the bottom left corner.
Benoît Trémoulinas (Sevilla) is shown the yellow card for hand ball.
Penalty conceded by Benoît Trémoulinas (Sevilla) with a hand ball in the penalty area.
Attempt blocked. Sandro Ramírez (Barcelona) right footed shot from outside the box is blocked. Assisted by Neymar.
Substitution, Sevilla. Fernando Llorente replaces Kevin Gameiro.
Attempt missed. Gerard Piqué (Barcelona) header from the centre of the box misses to the right. Assisted by Neymar with a cross following a corner.
Corner,  Barcelona. Conceded by Coke.
Barcelona lost to Sevilla as they suffered their second La Liga defeat of the season, despite Neymar's penalty.