MEPs have voted in favour of a resolution to allow the Brexit negotiations to progress to phase two.
The MEPs voted by 556 to 56 in favour of the resolution at the European Parliament in Strasbourg.
Before the vote, the EU's chief Brexit negotiator said there could be no going back on a deal agreed between the EU and the UK last week.
Mr Barnier told the European Parliament that the deal reached last week would be translated into a legally binding agreement.
"There is no going back on this sufficient progress, this progress has been noted and recorded and is going to be translated into a legally binding withdrawal agreement," he said.
"These are prerequisites and conditions for continuing with the negotiations in a smooth fashion."
Mr Barnier said Ireland would be the subject of a special negotiation path during phase two of the Brexit discussions.
"Everyone is going to have to work towards a specific solution for a unique situation," he said.
He said the European Commission would never allow concessions on the issues of citizens' rights and Northern Ireland.
Manfred Weber, leader of the EPP group in the European Parliament, said Ireland was stronger because of its membership of the European Union.
His comments were met with shouts of "rubbish" from UKIP MEPs sitting with Nigel Farage.
MEPs will also insist that the UK government gives a firm commitment to stick to the promises made last week, and formally criticise UK Brexit minister David Davis.
Mr Davis's weekend comments, suggesting the Brexit deal done last week was a statement of intent rather than a firm commitment, did not go down well in Strasbourg.
The European Parliament's Brexit negotiator Guy Verhofstadt told the parliament that the EU would stand behind Ireland during the negotiations.
"Ireland cannot become collateral damage of Brexit," he said.
He said there was a danger that the UK would leave the EU in name only, and that the next government would sign up to a customs union forever.
"The aspirations of the people are in a very different place, and Brexit may have to be refought over again," he added.
We will never allow the Irish people to become the collateral damage of #Brexit. We will continue to stand behind Ireland. #EPlenary
— Guy Verhofstadt (@guyverhofstadt) December 13, 2017
Mr Farage criticised British Prime Minister Theresa May for the deal last week.
Describing her as "Theresa the appeaser" he said she had danced to the EU’s tune.
