Diarmuid Martin was speaking after a Mass to close a two-day international conference on addiction hosted by the Pioneer Total Abstinence Association.
The Catholic prelate urged Irish society to address "the commercialisation of drink" and even the way "people talk about drinking – almost as if to normalise it".
He urged the Government to do more to support addiction treatment programmes which he described as "vital" but he said society in general needed to be much more involved too.
It came as the first letter from Pope Francis to an Irish-based organisation arrived in time for the Pioneer's conference at All Hallow's College in Dublin.
In it the Pontiff congratulated the Pioneers on their 115 years, "through which countless people have been assisted in living full lives without dependence on alcohol or drugs".
Meanwhile, Fr Peter McVerry told the 200 delegates that of the €3m needed to run his three addiction treatment programmes, only €190,000 was grant aided from the State. The rest had to be fundraised.
"I think that reflects the low priority that drug issues have in our society and with the Government," he said.
"We have probably 20,000 heroin users in Ireland today and we have just 28 beds in detox treatment programmes – that is a disgrace," Fr McVerry said.
Archbishop Martin meanwhile has also defended the Mater Hospital's pro life record saying it had been 'scrupulous' in trying to defend both the life of mother and the unborn child.
The Archbishop was also asked about the decision by the Mater, a Catholic Voluntary Hospital, to comply with the terms of the Protection of Life During Pregnancy Act.
He said that though he was president of the hospital he had no powers in the governance of the hospital. And he paid tribute to the Mater hospital's "great tradition of caring for very difficult pregnancies and doing it well within the ethos of the hospital over years".
"There have been extremely complicated (pregnancies) and I know that they are scrupulous in the policy of trying to defend both the life of the mother and the unborn child. I hope that that continues," he said.
He said he would be seeking further clarifications on the exact meaning of the hospital's statement last week.