The team manager said the players "apologise from the bottom of our hearts" after criticism online, and from politicians.
The Football Association of Ireland has apologised after a video emerged of players from the women's team appearing to chant in support of an illegal terrorist organisation.
Videos on social media showed the team celebrating in their changing room after a 0-1 win against Scotland in Glasgow on Tuesday night.
The victory secured the Republic of Ireland's spot at the 2023 Women's World Cup, to be staged in Australia and New Zealand.
A number of footballers appeared to chant "Ooh ah, up the Ra," an apparent reference to the outlawed Irish Republican Army, a Northern Irish terror group responsible for the deaths of hundreds of civilians, police, and military personnel during a decades-long campaign against British rule in the province.
“We apologise from the bottom of our hearts to anyone who has been offended by the content of the post-match celebrations after we had just qualified for the World Cup,” said team manager Vera Pauw, who also apologised "for any offence caused by a song sung by players in the Ireland dressing room".
Pauw said she would be reviewing the incident with the players, "and remind them of their responsibilities in this regard".
"We are collectively sorry for any hurt caused, there can be no excuse for that."
The leader of the pro-British Ulster Unionist Party Doug Beattie said that if the clip was accurate "it's a kick in the teeth for some of those who congratulated them, and those supporters who lost loved ones at the hands of the IRA".
Another Northern Ireland politician, Diane Forsyth from the Democratic Unionist Party, condemned the "disgraceful behaviour" of the players.
Forsyth said she would be writing to the Football Association of Ireland and World Cup organisers asking what steps they will take.
"This was a barbaric terrorist organisation. To be singing such a song is utterly insensitive to Provisional IRA victims," she wrote in a statement.