Mr Davis had previously said not settling both elements simultaneously would be "the row of the summer".
Chancellor of the Exchequer Philip Hammond said on Friday that protecting jobs and the British economy would be priorities for his country.
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.
The two chief negotiators, Michel Barnier of the European Union and David Davis from Britain, immediately set off to find common ground in their working relationship, an important touchstone to see how amicable the biggest political divorce in decades will become.
Though May's gamble will have done little to strengthen the U.K.'s position in European Union talks, analysts suspect it will do little to change its intended Brexit agenda.
A look at what is ahead now that Brexit talks have started
So, we each have to assume our responsibility and the consequences of our decisions. "The consequences are substantial".
The talks will focus on issues regarding citizens rights, the United Kingdom.
The mood was "incredibly positive" on the first day of Brexit talks between the two sides and they acknowledged that it was time to move quickly, a British source said.
He vowed to seek "a deal that works in the best interests of all citizens" with Mr Barnier as the pair began their discussions at the commission's Berlaymont headquarters in the Belgian capital.
His priority, he said, was to clear up the uncertainties which last June's Brexit vote had created.
Mr Barnier quoted the founder of the trading bloc that later became the EU, Jean Monnet, as saying: "I am neither optimistic nor pessimistic".
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Mr Davis, looking slightly more windswept than his flinty European colleague after the day's talks, brushed off the idea Britain's negotiating stance could change given political instability in the United Kingdom, with Prime Minister Theresa May seeking support of the Democratic Unionist Party (DUP) to prop up her minority government. "Our aim is to have one week of negotiations every month", said Barnier that further described the break-down of talks to into three groups that are to deal with: Citizens rights, Single financial settlement and Other separation issues.
A terms of reference document setting out the rules for both sides also states: "For both parties the default is transparency".
He added: "Obviously this is the first day of the talks on Brexit and I think the most important thing is we should all start - of course there'll be lots of discussions about the nature of the deal we are going to do - but I think we should also enter on the discussion about money and so and so forth. We must first tackle the uncertainties caused by Brexit", centering on citizens living on each other's territory, border arrangements between Ireland and the United Kingdom and the amount that Britain stands to pay to get out of its previous European Union commitments, Barnier said.
Other issues on the agenda, at least in the near term, include establishing residential rights of more than 3 million European Union citizens now living in the United Kingdom - and the 1 million British expats living in European Union member states - once the formal exit is complete.
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.
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