Finance Minister Michael Noonan has said the Irish Government will appeal the ruling.
The Irish Government has said it "disagrees profoundly with the commission's analysis".
"Ireland did not give favourable tax treatment to Apple," it added.
"Ireland does not do deals with taxpayers. No fine or penalty has been levied against the Irish State.
"This decision has no effect on the 12.5% rate of corporation tax and is not about Ireland's wider corporation tax regime."
Office 'existed on paper'?
The European Commission said Ireland's tax rulings had allowed Apple to pay substantially less tax than other businesses.
The Irish system had allowed profits to be attributed to a head office that "only existed on paper", said Margrethe Vestager, the European Commissioner for Competition.
Thirteen billion euros is approximately equivalent to a quarter of what the Irish government spends per year, and what it spends on health in total.
The Republic's government is a minority one, with Fine Gael being propped up by independents.
It is not clear whether the Independent Alliance cabinet members will vote to appeal the decision.
Apple itself is appealing against the decision, saying: "The European Commission has launched an effort to rewrite Apple's history in Europe, ignore Ireland's tax laws and upend the international tax system in the process.
"The commission's case is not about how much Apple pays in taxes, it's about which government collects the money. It will have a profound and harmful effect on investment and job creation in Europe.
"Apple follows the law and pays all of the taxes we owe wherever we operate. We will appeal and we are confident the decision will be overturned."
The appeal will be made first to the General Court of the European Union in Luxembourg and then to European Court of Justice.
Fianna Fáil's finance spokesman Michael McGrath described the Europe Commission's decision as "remarkable", saying it would have far-reaching consequences.
But many of the issues included by the commission have been addressed through changes in legislation in recent years, he said.
It was vital from Ireland's perspective that the country's corporation tax system, which is an essential part of the country's overall inward investment strategy, was defended, he added.
Sinn Féin's Finance spokesperson, Pearse Doherty, has called for a public inquiry into what he described as "the illegal State aid given to Apple."
He said that his party would be tabling a motion in the Dáil calling on the Government not to appeal the European Commission's decision.
Labour leader Brendan Howlin called on Mr Noonan to brief the Opposition on the matter but added that Ireland should appeal the decision along with other States.
Richard Boyd Barrett of AAA-PBP said the Government should pursue Apple for the money which would help resolve the housing crisis.
Apple chief executive Tim Cook has said the company is committed to Ireland and plans to continue investing.
Mr Cook has published message on the company's website reiterating the organisation's commitment to Ireland.
"We plan to continue investing there, growing and serving our customers with the same level of passion and commitment.
He added: "we firmly believe that the facts and the established legal principles upon which the EU was founded will ultimately prevail".
Apple and Ireland
€13bn demanded in back taxes. Equal to:
- ALL of Ireland's healthcare budget
- 66% of its social welfare bill
- 15 million iPhones
- 27% of Apple's 2015 profit
Sources: Apple, HSE, Dept of Social Protection
Analysis: BBC Ireland Correspondent Shane Harrison
It all comes down to Ireland's reputation.
The Irish government has been criticised, particularly in the US, for almost being a tax haven similar to the Cayman Islands, something it strongly denies.
But €13bn is a huge amount of money: It would go a long way towards solving the housing crisis in the Republic for example.
Fine Gael, Fianna Fáil and Labour are more or less of the view that the government should appeal the ruling.
But Sinn Féin are calling for an independent assessment of whether or not Apple did get unfair treatment, whether it was human error or whether the system colluded to create this situation as the European Commission seems to believe.
It believes the government should not appeal and should spend the money.
The left-wing parties also believe the money should be spend in citizens' interests.
For them, it is a moral issue as to whether or not Apple are being getting unfair treatment given the amount of money ordinary people pay in their taxes.