Players and staff from both teams rushed onto the field after a scuffle between Fenerbahce's Kerem Akturkoglu and Galatasaray's Roland Sallai, according to video shared on social media.
"The rivalry and tension of the derby do not give anyone the right to attack our players," Fenerbahce said in a statement.
"What happened cannot be dismissed as 'derby tension.' The behaviour of the opposing team's members, instead of reducing tension, fuelled the incidents."
Fenerbahce have called for an investigation and disciplinary action against those responsible for the brawl.