Reflecting the sensitivity of the topic, questions over Northern Ireland are to be handled in a "dialogue" between deputy negotiators, which London hopes will open the way to discussing matters relating to future trade.
David Davis began negotiations yesterday.
"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.
The aim is to make sufficient progress to get leaders of the other 27 European Union countries to agree to move on to talks on a future relationship with Britain, including a trade deal, he said.
The June 2016 Brexit vote in the hotly contested referendum exposed deep division across the country.
The UK is set to leave the European Union by the end of March 2019.More news: Germany up against Mexico; Chile to face Portugal
Brexit Secretary David Davis appears to have conceded to European Union demands during the first day of historic talks, by agreeing to a "two-phased" negotiation.
Brexit negotiations begin after almost a whole year of speculation and build up. "I am working with the British Government and its official representative David Davis".
Barnier said the two sides will have one week of negotiation every month, and use the time in between to work on proposals and exchange them. These facts are signs that she might face pressure to a secure a soft Brexit deal as opposed to her original plan of a hard Brexit which would have meant she would have more control over immigration and law-making. "We are talking about orderly withdrawal first, and that makes sense".
Three negotiating groups will address agreed priorities on citizens' rights, finances and other technical matters which must be resolved to avoid a messy limbo after Brexit.
He has also held talks with the EU Council President Donald Tusk.
"Then in the second step, we scope our future partnership; we also agree on how we structure our tools".More news: Camilla Tell-All says Princess Diana was paranoid after Charles' affair
Following the first day talk Barnier said, "It was very important and useful to start off on the right foot as the clock is ticking".
Although the United Kingdom initially showed resilience against Brexit risk, a weaker sterling has driven up the prices of goods, with inflation hitting 2.9% in May, leading to less consumer spending.
He also stuck to May's position that Britain will pursue a completely new trade agreement with the union.
"It's not when it starts it's how it finishes that matters", he said. "In a first step, we will deal with the most pressing issues", Barnier said.
Asked if the European Union had made any concession to the UK, Barnier said: "The UK has chose to leave the EU". It's not about punishment, it is not about revenge.
"My mandate is to defend the single market, defend the European Union, which the United Kingdom has chose to leave", Barnier told reporters after briefing ministers from EU member states in Luxembourg.More news: Arsenal's £31m bid for Thomas Lemar turned down by Monaco
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