A group of between 40 and 50 hooded fans clashed before kick-off, although the police added no arrests have been made.
The injured supporter has been taken to hospital suffering from head injuries.
Both sides released statements condemning the violence.
"FC Barcelona regrets and condemns the incidents that occurred this afternoon in the city of Vitoria, before the match against Deportivo Alaves," read Barcelona's statement.
"The club strongly rejects any violent conduct and hopes that the authorities will be able to identify those responsible for these acts as soon as possible."
Videos released in the Spanish media showed fans throwing chairs and wielding metal bars.
"Deportivo Alaves profoundly condemns the violent acts that took place this morning February 11, in the university zone between supposed fans of Alaves and FC Barcelona," Alaves said in their statement.
"The club will put itself at the disposition of the authorities and, if it is confirmed that those implicated are season ticket holders, Deportivo Alaves will consider the necessary sanctions to take."
The match ended in a 6-0 Barcelona win.
The two sides are due to meet again in the Copa del Rey final on May 27.