The two sides agreed to set up three working groups covering the details of withdrawal: the financial settlement; citizens' rights; and one dealing with other legal issues relating to "separation".
In choreographed talks that saw the two men exchange mountaineering gifts, they agreed to discuss divorce issues before negotiations on a future trade deal can start.
He hailed that the first session was "important", "open" and "useful indeed to start off on the right foot as the clock is ticking", Xinhua news reported.
"The United Kingdom is going to leave the European Union, single market and the customs union, not the other way around".
Mr. Barnier has said he hopes to conclude the talks by October 2018, leaving six months to ratify the deal. "And the consequences are substantial". Last week, one diplomat said May had tried to "hijack" the summit taking place on Thursday and Friday by drawing other leaders into wider discussions on Brexit.
He said May will publish a detailed paper outlining Britain's approach on protecting citizen rights at the European Union summit on Thursday and Friday, which he believes "will form the right basis on which to reach agreement".
"There will be no hostility on my side, I will display a constructive attitude firmly based on the interests and support of the 27".
The EU said it was also looking for a good compromise.
"Let's wait for that paper. there may be some common ground".
He said those were the rights of expatriate citizens and problems of a new EU-UK border, notably cutting across Ireland.
"What we also need is certainty, for our companies in Belgium, in Europe", he said. We have taken the first, critical steps together.
"Naturally we will conduct these negotiations quickly and we will conduct them intensively".
United Kingdom concession comes after just one day of talks with European Union chief negotiator Michel Barnier.
On Monday, talks over the UK's withdrawal from the bloc officially began, and are due to be completed by March 30, 2019.
In a joint press conference with his British counterpart David Davis, Barnier said that they have agreed on the dates and on the priorities of the talks.
"The best start would be to abandon the unsafe idea that we could leave the European Union with no deal at all, which would be the worst possible outcome for jobs and living standards in this country".