The victory saw Barca pull further clear of Real Madrid at the top of the early La Liga standings, and after the match Suarez paid tribute to "the best in the world".
By the hour mark, the Argentine captain had made it 3-0.
From there the floodgates opened, and although Messi should have been flagged offside, no call came as he controlled Ivan Rakitic's pass before cleverly turning David Lopez and rifling into the top corner.
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.
He said: "We want to do something important for our people and we hope to do our best because it will be demanding".
As Messi looked to dart into the left side of the penalty box, the ball cannoned up off an Espanyol defender, before then being hit against team-mate Pape Diop.
It only took 10 minutes for Messi to bag his second, thanks to an assist from Jordi Alba.
Uruguay forward Suarez, who only returned from worldwide duty on Friday, looked to be having one of those nights as he sent a cut-back from Messi over from six yards.
Despite Barcelona's positive start, Valverde wants to keep things in perspective as they look to maintain an advantage over champions Real Madrid.
Valverde may also turn to Brazilian center-back Paulinho, who has played two matches for his country's qualifying campaign for the 2018 World Cup in Russian Federation, where he will face Espanyol on the bench to prepare for Juventus in the Champions League.