The new Metro North Line should be in operation by 2026/27.
It is part of a 27bn euro (Â£20bn) plan for funding on capital projects in the Republic of Ireland from 2016 to 2021.
The plan should lead to 45,000 new construction jobs and a series of new road and schools projects across the Republic of Ireland.
The company that owns Dublin and Cork airports welcomed the news.
"We have been highly supportive of the plans for a rail link from Dublin Airport to the city centre for many years and we're pleased that the government has approved the new Metro North option," said Dublin Airport Authority chief executive Kevin Toland.
"A rail link to the city centre is an essential element of Dublin Airport's long-term growth plans and we welcome the fact that the new Metro service will be underground at the airport. This will allow an efficient transfer of passengers from the Metro to the terminals and vice versa."
Mr Toland said the transfer time from O'Connell Street, Dublin to the airport would be about 19 minutes.
The Irish government has announced plans to build a new underground metro rail link from the centre of Dublin to the city's airport.