Denmark international Krohn-Dehli tucked the ball home after 52 minutes following a pass from Kevin Gameiro to give Sevilla the lead and then provided a cross for Vicente Iborra to nod in six minutes later.
Barca, with Lionel Messi absent through injury, hit the woodwork on four occasions but luck was not on their side as they lost their second consecutive away game following their 4-1 defeat by Celta Vigo.
For Sevilla, victory was some revenge for their European Super Cup loss in August.
Neymar struck the post with a free-kick after 23 minutes, with the ball then striking the woodwork again after rebounding off keeper Sergio Rico, and running across the goal line just out of the reach of Gerard Pique.
Luis Suarez also fired against the crossbar with a curling effort from outside the area before half time but Barca’s defending was poor and they paid the price after the break.
Neymar scored from the penalty spot on his 100th appearance after a 74th minute handball from Benoit Tremoulinas and then substitute Sandro Ramirez was denied by the woodwork with 10 minutes to go.
Barca have 15 points from seven games, a point behind leaders Villarreal who are away to Levante on Sunday.
Fourth-placed Real Madrid, on 14 points, take on Atletico Madrid who are one place and two points below them also on Sunday.
SEVERAL CHANCES
"It is frustrating to lose like that as we had several chances and they scored with the first two they had," Barca midfielder Ivan Rakitic told reporters.
"We now have to lift ourselves and go for everything in the next few games."
Barca’s injury list also included Andres Iniesta but Rakitic did not blame the absentees.
"This is part of football. It is tough with players out and we hope they come back quickly but it is the same for all teams."
Keeper Claudio Bravo was back for the Catalan side and replaced under-fire Marc-Andre Ter Stegen who has been criticised for some recent displays, but he could do little with the goals.
Sevilla stretched the Barca defence from the start with Gameiro in particular a threat but it was Suarez who had the first clear chance.
The Uruguayan was slightly slow to get his shot away inside the area after 13 minutes and it was blocked by Timothee Kolodziejczak.
Elsewhere, Charles grabbed a hat-trick for Malaga in a 3-1 victory over Real Sociedad who scored through Imanol Agirretxe.
Piti's goal gave Granada a 1-1 draw at home to Deportivo La Coruna after Faycal Fajr had given the visitors the lead.
Saul Berjon and Borja Baston struck for Eibar in a 2-0 triumph at Las Palmas while Alex Menendez's injury-time winner helped Sporting Gijon beat Espanyol 2-1.
(Editing by Alan Baldwin)
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