Zlatan Ibrahimovic poked the La Liga champions in front in the 20th minute of Spain's traditional season opener at Sevilla's Sanchez Pizjaun stadium, before Fabiano levelled for the King's Cup winners just after the hour.
Substitute Kanoute fired the hosts in front with just over 15 minutes left and nodded a second in the 83rd to put the Andalusians in a strong position ahead of the second leg next Saturday at the Nou Camp.
Barca coach Pep Guardiola opted not to take any of the eight members of Spain's World Cup-winning squad to Sevilla, also leaving World Player of the Year Lionel Messi on the bench and including several youth-team players in his starting line-up.
Local media have speculated Ibrahimovic will leave Barca before the transfer window closes at the end of the month but Guardiola deployed the burly Sweden striker at centre forward on a sweltering night in Seville.
He went some way toward silencing his critics when he sprinted on to Maxwell's curled pass and slipped the ball low past goalkeeper Andres Palop.
Sevilla looked much more dangerous after the break and had a deserved equaliser when Fabiano ran on to a pass from new signing Luca Cigarini and slotted through the legs of Barca goalkeeper Mino.
The Brazilian striker, who Olympique Marseille are attempting to lure to Ligue 1, pointedly grabbed the Sevilla shield on his shirt as he celebrated.
Kanoute flicked the ball high into the net from Alvaro Negredo's cross to put Sevilla ahead before heading powerfully home from Diego Perotti's cross as Barca wilted.
"We suffered a bit at the start but we played a very good second half," Kanoute said in a television interview.
"But they were missing a lot of players and we'll have to be very careful in the second leg," the Mali striker added.
