The 21-year-old finished after being picked out by Mikel Merino midway through the first half, and the strike proved enough for the Basque side to claim all three points.
Kubo moved to San Sebastian last month on a five-year deal from Real Madrid, after never managing to break into the first team at the Santiago Bernabeu.
Signed by Madrid from FC Tokyo as an 18-year-old in 2019, he was farmed out on loan to Mallorca, Villarreal and then Getafe, before returning to Mallorca last season.
La Real finished sixth in La Liga in the last campaign and will play in the group stage of this season’s Europa League.
Reigning champions Real Madrid begin their defence of the title away at newly-promoted Almeria later.
Barcelona were held to a 0-0 draw by Rayo Vallecano in their opening game at the Camp Nou on Saturday.