The league's disciplinary commissioner did not name the player involved. Milan's former Genoa player and Guinea international Kevin Constant was sent off in the game.
Milan's Mario Balotelli and Sulley Muntari were racially abused by Inter Milan supporters last month. Team-mate Kevin-Prince Boateng stormed off the pitch when racially abused by Pro Patria fans in a friendly in January.
The Serie A statement said Genoa's fine was also because of lasers being shone by home fans at Milan players and coins and lighters being thrown at a goal-line official and Milan goalkeeper Christian Abbiati.
Lazio were fined 10,000 euros after fans unveiled a banner insulting UEFA President Michel Platini during Sunday's 2-0 home defeat by Fiorentina.
Last month UEFA ordered Lazio to play two European games behind closed doors because of fan racism.
An appeal by the Rome club had failed, president Claudio Lotito told reporters on Monday.
