Gerard Pique's header made it 4-0 with three minutes left before £96.8million signing Dembele, on as a late substitute, laid a fifth on a plate for Suarez in stoppage time.
Lionel Messi's hat-trick had Barca comfortably in the lead by the time the 20-year-old, who joined the club for €105 million late in the summer transfer window, got on the pitch, with the Argentina global netting his third just moments before the youngster was brought on in the 68th minute.
Meanwhile, not too far away from there, Real Madrid drew, at home, for the second straight match.
Ernesto Valverde described the Argentine as'extraordinary following the final whistle
The visitors were initially successful in that aim, with their disciplined defending preventing the hosts from creating any chances in the opening 20 minutes other than a free-kick from Suarez which Pau Lopez safely got behind.
Messi scored the first goal in the 26th minute when he evaded two markers to unleash a shot from close range.
With both Real Madrid and Atletico having dropped points with draws earlier in the day, Barcelona were looking to strike an early blow at the top of the La Liga table.
New royal baby to shake up House of Windsor pecking order
George will be dressed in his new uniform, a navy v-neck pullover, matching Bermuda-style shorts, long red socks and black shoes. The Duchess suspended all Royal engagements as a result of the illness, for which she has previously been hospitalised.
Barcelona made all the early running but suffered a scare in the 21st minute when Sergio Busquets booked for cynically pulling back Leo Baptistao after his misplaced pass threatened to give the Espanyol forward a route to goal.
However, replays showed Messi had been at least a yard offside as the Espanyol back line stepped up.
Ganso opened the scoring in the 46th minute before Wassim Ben Yedder doubled their lead in the 76th minute and Nolito sealed an eventually comfortable win in injury time.
Woodburn shines for Wales again
I know he won't get too carried away - I'm sure if he does we'll all let him know! "He's been great since he came in". Global teammate Gareth Bale had only positives to say about Woodburn. "Ben's done that and we are all proud of him".
The Argentina forward completed his 38th Barcelona hat-trick when he finished off from another Alba cross.
Espanyol almost pulled a goal back just before the break when Pablo Piatti rattled the upright from the edge of the box.
Suarez who shrugged off an injury whilst on worldwide break missed some decent chances, the closest being in the 55th minute when he blazed wide in the six yard box after a cutback from Messi.
Stoke ends United's ideal start to EPL season
Stoke: Butland 7, Cameron 6, Zouma 8, Wimmer 6, Diouf 6, Fletcher 6, Allen 6, Pieters 6, Shaqiri 7, Jese 6, Choupo-Moting 8. When questioned about it later, Hughes simply laughed it off but Mourinho snapped at the reporter who asked the question.
Next up for Barcelona is their Champions League group opener against Italian champions Juventus on Tuesday night, while Espanyol will continue their La Liga campaign at home to Celta Vigo on September 18.
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