A video posted on social media after the 1-0 playoff second leg win in Glasgow showed players chanting a slogan supportive of the paramilitary Irish Republican Army that sought to end British rule in Northern Ireland.
UEFA's control, ethics and disciplinary body said in a statement it had fined the FAI for a "violation of the basic rules of decent conduct".
Ireland manager Vera Pauw (59) and the FAI had apologised for the incident.
