Kanoute scored the opening goal in the eighth minute from a tight angle, and helped make it 2-1 in the 27th with a back-heel pass to Luis Fabiano, who shrugged off his defender before rounding goalkeeper Martin Vaniak and slotting it into the empty goal.
Daniel Pudil had equalized in the 19th for Slavia.
Julien Escude pushed Seydour Keita's header from a corner kick into the roof of the goal in the 58th, and Kanoute fed substitute Arouna Kone in the 69th.
Substitute David Kalivoda slotted in a rebound for Slavia Prague in the second minute of injury time.
Sevilla and Slavia Prague both have three points from two games. Arsenal leads Group H with six points after beating winless Steaua Bucharest 1-0.
Sevilla came into the first European Cup match at the Ramon Sanchez Pizjuan for 50 years after three straight league losses and a 3-0 defeat at Arsenal.
But in steady rain, Kanoute and Fabiano troubled Slavia's defense all night and created chances from the start.
Fabiano missed with a left-footed shot before Vaniak jumped to block Keita's hard shot from the edge of the area within the first six minutes.
Kanoute opened the scoring after rounding the charging Vaniak.
Pudil took advantage of a defensive lull, latching on to a bouncing ball to send a left-footed volley past diving goalkeeper Andres Palop into the top left of Sevilla's goal.
Fabiano missed with a header over the crossbar and Kanoute shot into the side netting from a tight angle before Vaniak dived to stop Fabiano's low shot at the right post.
Sevilla sluggish defense gave Tijani Belaid the chance for a shot just wide in the 53rd, but the home side responded quickly with Escude's goal.
Vaniak saved a header from Keita and Kanoute shot just wide after a pass from Kone, who replaced Fabiano.
Kone then curled Kanoute's short ball around Vaniak inside the post to make it 4-1 before Kalivoda pounced on a late rebound for Slavia's second.