Griezmann's place in the newly-formed attacking trident has not been secure this season but the Frenchman opened the scoring at Ipurua before teeing up Messi and then playing the crucial pass for Suarez to make it three, AFP reported.
An authoritative Barca display dismissed suggestions the players might be affected by Friday's postponement of the Clasico due to protests in Catalonia.
Violence in the city of Barcelona and disruptions at El Prat Airport meant Ernesto Valverde's side chose to alter its pre-match routine and embark on a 589-kilometer bus trip to the Basque County on Friday.
And all the while, discussions continued over the Clasico against Real Madrid, which had originally been scheduled for next Saturday at Camp Nou but was postponed on Friday and is expected now to go ahead in December.
Perhaps chaos off the pitch inspired Barca to unite on it.
This was its fifth consecutive victory in all competitions, a sequence which is swiftly erasing memories of Barca’s stumbling start to the season.
The resurgence has coincided with Messi's return from injury but it was the fluidity and cohesion of the front three, rather than the Argentinian's familiar solo brilliance, that will have been most pleasing for Valverde.
Four for Griezmann
Griezmann now has four goals since joining Barca from Atletico Madrid last summer but he has not entirely convinced so far, a series of underwhelming showings leading to Ousmane Dembele taking his starting spot against Sevilla before the international break.
Dembele might even have continued had he not been sent off and suspended for this game but Griezmann certainly took his chance. This was the third time he has started alongside Messi and Suarez but the first that they have really gelled as a trio.
There was good news too for Barcelona in defense, where Samuel Umtiti came through 90 minutes in what was his first appearance of the season. Umtiti, who has been out with a foot injury, replaced the banned Gerard Pique.
The only blot on the afternoon was an injury to Sergi Roberto, who hurt his left knee and was replaced at halftime.
The visiting fans chanted in support of the nine Catalan leaders imprisoned last week for their failed independence bid in 2017.