The Argentine superstar completed his hat-trick with a clinical finish in the 67th minute.
For all Barca's off-field problems, Ernesto Valverde's men have taken the maximum points from their opening three games.
Messi opened the scoring at Camp Nou when he received a pass from Ivan Rakitic in what appeared to be an offside position.
Espanyol were being outplayed but retained a sporadic threat, with a combination of Pique and Samuel Umtiti diverting Baptistao's 61st-minute shot behind.
Suarez was back in the Barcelona line-up after an injury scare while on worldwide duty with Uruguay, but was not enjoying himself in front of goal as much as Messi, fluffing shot from a Gerard Deulofeu's cross before being denied by Pau early in the second period.
However, no matter how much interest Dembele and Paulinho may provoke, Barcelona President Josep Maria Bartomeu is likely to have nearly as many camera lenses focused on him as the players.
"We can't be happy with our performance, above all in the first half", said Real boss Zinedine Zidane.
With a squad that looks weaker than last season and most importantly, weaker than what Real Madrid have to offer, this season was always going to be about Messi, first and foremost. "We need to have patience and think about the next game, which is just around the corner".
Their title rivals Real Madrid and Atletico dropped points with draws earlier in the day. Goals from Paulo Henrique Ganso, Wissam Ben Yedder and Nolito secured the win for Sevilla.