Suarez had a golden chance to level the scores late in the first period when he found space inside the box, but the Uruguay worldwide missed the far post as Sevilla kept their one-goal lead in tact.
The 21-year-old winger was signed in a surprise £36.5 million deal from Bordeaux this summer and was expected to make his official debut against Sevilla.
While Barcelona dominated possession, Sevilla went close to going back in front when Franco Vazquez's glancing header hit the top of the goal-frame in the 62nd before Dembele took his victor.
Barcelona goalkeeper Marc-Andre ter Stegen secured the win when he saved a penalty kick from Sevilla substitute Wissam Ben Yedder in the 89th after Ter Stegen fouled former Barcelona forward Aleix Vidal.More news: Here's how and when to watch the Perseid meteor shower
More news: Oscars set to change
More news: Brazilian Ronaldo In Intensive Care In Ibiza Hospital
It was Barcelona's 13th Supercup and sixth in the last nine seasons.
And while Messi may need a few more matches to drop back into his scintillating regular form, it would take a courageous man to bet against the 31-year-old star adding to that collection by the end of the current season. "He has an fantastic ability to break down defences with his speed".
The Spanish Super Cup preceeds every La Liga opener and the match it is between La Liga champions against the Copa del Rey winners, but since Barcelona won the domestic double last season, Sevilla qualify as Copa del Rey runners-up.
Barcelona's new summer signing Malcom has been left out of the Spanish Super Cup against Sevilla on Sunday evening.
The victor of the match that will be staged in Tangier, Morocco, will be decided over one leg for the first time. Sevilla faces Zalgiris on Thursday after winning their first match 1-0.