Earlier, Real paid for the absence of the suspended Cristiano Ronaldo and a series of missed chances by Gareth Bale as they were held to a shock 1-1 home draw by newly-promoted Levante.
Messi put Barcelona on their way with the opening goal after 26 minutes, although he looked to have been offside.
The hosts raced into a two-goal lead inside 35 minutes thanks to a brace from Messi.
Messi and Suarez played one-twos into the box and the ball broke to Alba, with the mercurial Argentinian on hand to convert the cutback once more to take the lead for three, with Dembele introduced shortly afterwards, followed by fellow new signing Paulinho shortly afterwards. But for the Brazil national team, not much changes.
A mix-up between Ter Stegen and Alba nearly gave Espanyol a lifeline in the 51st minute but Piatti's lofted shot bounced just wide. France forward Ousmane Dembele went on as a secondhalf substitute and set up Luis Suarez to complete the demolition of Espanyol.
Suarez who shrugged off an injury whilst on global 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.
Pique piled further misery before Dembele's helped Suarez end his misery, sealing the match right on the end of regular time.
A third win in as many rounds left Barcelona atop of the standings, with two more points than second-place Sevilla and four more points than defending champion Real Madrid in sixth. Both Madrid teams have five points.
With the result secure, Ernesto Valverde sent on Dembele in the 68th minute to make his debut, replacing Gerard Deulofeu to great applause from the Nou Camp faithful.