Messi broke the deadlock after 25 minutes as he collected a through-ball from Ivan Rakitic to fire home.
The hosts eventually bagged a third goal after 67 minutes as Messi completed his hat-trick with a composed finish, after a clever pass from Jordi Alba.
Gerard Pique's header made it 4-0 with three minutes left before Â£96.8million signing Dembele, on as a late substitute, laid a fifth on a plate for Suarez in stoppage time.
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.
Ernesto Valverde banks on debutant Ousmane Dembele to make instant impact against Espanyol
Suarez's trials continued as he looped an attempt on to the roof of the net having been set up by Messi - both forwards jogged back into position wearing rueful smiles. Gerard Pique headed in a fourth goal.
Sky Bet now make the Catalan giants, who are two points clear of second-placed Sevilla at the top of the table, 4/6 to win the title while champions Real Madrid have drifted to 5/4 after they dropped points for the second game in a row.
- Messi has scored two or more goals in 10 of his last 15 LaLiga games - he didn't score in the other five.
So fond are Espanyol of their former striker that he has a gate named after him at their stadium but on Saturday they will be hoping to take advantage of institutional instability at Barcelona to end their torrid run at the Nou Camp.
On the pitch Barcelona have won their opening two games of the season, despite not playing their best football and they can expect Espanyol to again make things hard for them.
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It had also, the Commission found , paid German retailer Media-Saturn for selling exclusively x86 computers with Intel processors. In June the Commission hit Google with a €2.4bn fine over its online shopping services.