Messi opened scoring in the 26th minute amid suspicion of offside and doubled the lead 10 minutes before the break. The Argentine found Suarez who laid a pass onto the path of Jordi Alba.
The 20-year-old, who signed for £135m from Borussia Dortmund this summer, came on a substitute in the 68th minute with Barca already 3-0 up thanks to a hat-trick from Lionel Messi.
Things are looking rosy at the Camp Nou.
Barca remain top of the table with nine points.
Kohli fires India to seven-wicket T20 win
When India were 2 down, Sri Lanka were right back in the game with scoring rate being brought down and new batsmen at the crease. The focus of the lone T20I was whether India would be able to complete a 9-0 thrashing of Sri Lanka across formats.
Valverde, speaking about Messi's impact for the team in their match with Espanyol said in a post-match news conference: "I am happy for Leo who scored three goals and is extraordinary".
"We pulled through a complicated game, because these kinds of matches are always tough".
'Despite the result, it wasn't easy.
"It is a privilege to play with the best in the world, he continues to show that people have a long way to catch him", Suarez said.
Kerala will eat beef, no food emergency in India, says KJ Alphons
The new inductions include four former civil servants with the 1979 batch, Kerala cadre officer being one of them. He resigned from Indian Administrative Service to contest in 2006 Kerala Assembly elections.
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.
Pau Lopez made a fine save to deny Rakitic after he raced clear before Barcelona did go 2-0 ahead in the 35th minute, with another fortuitous goal.
Barcelona coach Ernesto Valverde will be banking on new signing Ousmane Dembele to fill the gap left by Neymar's departure as he looks to overcome off-pitch difficulties when the Catalans take on Espanyol on Saturday.
"It is not he was not quite on the ball (during his first sessions with us), but he needed some time to adapt to the circumstances, to know his team-mates and understand the way we play".
Northern California wildfires force evacuations
The fire that started Wednesday night for unknown reasons is burning out of control in extreme heat and very dry conditions. Butte County has set up an evacuation center at the Church of the Nazarene in Oroville, 2238 Monte Vista Ave.