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While €120 million ($194 million) signing was among the substitutes, Suarez's acrobatic overhead kick got the LaLiga champion off the mark, Arturo Vidal polished off a sublime passing move with the second and Dembele scored a solo effort to all but end the contest.
Lionel Messi wrapped up the win with a curling second-half free-kick and the synergy between Barcelona's front three, which has not always clicked with Griezmann involved, will not have been lost on Valverde.
The match ended in unsavoury fashion for Barca, though, debutant Ronald Araujo and Dembele sent off shortly before stoppage-time.
Luuk de Jong received good service from Lucas Ocampos in the early stages but he fluffed three chances put Sevilla ahead before Suarez broke the deadlock with a moment of magic 27 minutes in.
The Uruguay marksman watched Nelson Semedo's cross loop through the air and met it with an overhead kick into the bottom corner of the net before goalkeeper Tomas Vaclik could react.
The goal settled the hosts and five minutes later a move that incorporated 25 passes and all 11 Barcelona players culminated in Arthur's weighted ball, which allowed Vidal to slide in and glance a shot past Vaclik.
Dembele made it 3-0 inside eight first-half minutes when his slaloming run tore through Sevilla's defence, the 22-year-old opening his body and side-footing in at the far post.
The second half started just like the first, with De Jong squandering a great chance to score when he drilled a shot against the post from just inside the Barca box.
The hosts were soon back on top and Messi twice surged through Sevilla's midfield before putting the finishing touches on a fine win with a dipping, curling free-kick 12 minutes from time.
Valverde sent on Araujo for his debut, which turned sour when he was judged to have fouled Javier Hernandez as the Sevilla forward went through on goal and was shown a red card along with Dembele, who was sent off for his protests.
