Mr Flanagan was speaking at a conference attended by British and Irish politicians at Oxford.
Mr Flanagan also said that Northern Ireland's legacy inquest delays would be addressed by implementing the Lord Chief Justice's proposals.
The Lord Chief Justice, Sir Declan Morgan, requested Â£10m funding for a five-year programme to deal with inquests into some of the most controversial killings of the Troubles.
Mr Flanagan said: "I believe it is the solemn responsibility of politicians in London, in Belfast and in Dublin to deliver a framework for dealing with legacy issues."
Minister Flanagan also welcomed the new Secretary of State for Northern Ireland's stance on addressing legacy issues.
James Brokenshire said on Friday that he wants the search for agreement on the stumbling blocks regarding the legacy of the Troubles to enter a more public phase.
The Irish Foreign Affairs Minister Charlie Flanagan has said it is vital that other European Union member states understand the unique circumstances in the island of Ireland in the period before the UK triggers its departure from the EU.