Alvaro Garcia struck the only goal of the game after just seven minutes, with Barcelona struggling to build attacking momentum during much of the contest.
Still, Barca hit the woodwork toward the end of both halves, while Gavi had a strong penalty appeal rejected as the clock neared the 90-minute mark.
Xavi Hernandez's side will still be looking somewhat nervously over their shoulders with fifth-placed Real Betis six points behind. Only the top four teams qualify for next season's Champions League.