Monday's talks however are likely to focus on the practical details of timings for the coming months, with the big, divisive issues left aside for now, officials said.
Davis said the talks were off to a "promising start" and denied that Britain had caved in on the sequencing of the talks.
In a sign of the progress that has been made, Mr Davis said the Prime Minister would brief fellow European Union leaders at a summit on Thursday on the UK's approach to the rights of expatriate citizens, which will be set out in detail in a paper on Monday.
Britain on Monday finally opened negotiations with other European Union nations about leaving the bloc.
"We want to preserve all the aspects and all the obligations under the Good Friday agreement".
"I will display a constructive attitude firmly based on the interests and support of the 27 [member states]". Britain hopes the two themes - divorce terms and future relationship - can be discussed in parallel.
As for the first negotiations round, "talks about talks" have already bared fruit with Britain and the European Union agreeing on the priorities and timetable for Brexit negotiations after the first session of talks.
Working groups will discuss aspects of the negotiations in the afternoon before a final meeting between Mr Davis and Mr Barnier. The organisations collectively represent hundreds of thousands of businesses across the United Kingdom, which employ millions of people. Such firms sold 7.3 billion euros ($8.2 billion) worth of goods to customers in Britain previous year, their fourth-biggest market.
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Brexit negotiations which could define the UK's political and economic future have begun, with David Davis calling for a "new deep and special partnership" between Britain and Brussels.
Responding to Mr Barnier, Mr Davis quoted wartime British prime minister Winston Churchill: "No doubt the road ahead will at times be challenging, but as Churchill once said, a pessimist sees the difficulty in every opportunity".
Mr Barnier said it was important to tackle uncertainties caused by Brexit.
Davis, answering a question, said Britain's negotiating position had not changed as a result of his Conservative Party's poor showing in recent elections. EU leaders want May to lay off threats that she would walk out and leave a chaotic legal limbo for all Europeans. They should finally tell us what the aim is.
"Because membership of the single market requires the four freedoms [of people, goods, services and capital] to be abided, and [because] we need to bring back to Britain control of our laws, control of our borders, we'll be leaving the single market", he said.
"I look forward to beginning work on that new future".
"I think the whole process will lead to a happy resolution which can be done with honour and profit to both sides", Johnson said as he went into a separate meeting of EU Foreign Ministers in Luxembourg. "In the long run, this will be good for the United Kingdom and good for the rest of Europe".
The border between Northern Ireland, which is part of the United Kingdom, and the Republic of Ireland (Other OTC: IRLD - news), an European Union member, will become the only land border between the UK and the European Union after Britain exits the bloc in March 2019.
Germany's deputy foreign minister, Michael Roth, told RBB Inforadio that "we must of course protect our interests as the European Union 27 but naturally we also don't want to punish Britain". He added: "Brexit won't make anything better, but it will make a lot of things more hard".