The Croatian scored in the 21st moment when he landed at pace to control in Arda Turan's cross from the left.
Lionel Messi, Luis Suarez and Turan additionally went close for the champions.
Barca manager Marc-Andre ter Stegen verging on paid for a frightful pass, and Athletic's Raul Garcia and Inaki Williams went close for Bilbao.
Sunday's win sees mentor Luis Enrique's side join El Clasico equals Real Madrid and early pioneers Las Palmas on six focuses after two amusements of the new season.
The outcome ought to have been more decided for Barcelona, who squandered various possibilities.
Turan inspired in assault, yet maybe the standing memory of the Turkey midfielder's presentation was his first-half miss from eight yards.
Messi shot crawls wide after regularly floating past two protectors, furthermore discharged over the bar. Suarez, his accomplice in assault, thundered a volley over from eight yards furthermore had an exertion from two yards tidied up the line by Eneko Boveda in stoppage time.
Bilbao will lament missing the couple of good risks that came their direction. Midfielder Raul Garcia saw his strike barely float past Ter Stegen's post and Inaki Williams' jabbed exertion was spared by the Barca manager.
Ter Stegen was likewise lucky not to be left red-confronted in the principal half when his go out of protection was let go straight back at him by Spanish midfielder Benat.
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