"Today we agreed on dates, we agreed on organisation and we agreed on priorities for negotiations", he said after the talks at the European Commission in Brussels. "I'm well aware that throughout the process there will be points of compromise".
The EU said it was also looking for a good compromise.
European Union officials and diplomats greeted the agreement of Brexit Secretary Davis to a format and sequencing of talks that had been proposed by Barnier as sign that British Prime Minister Theresa May's weakened and potentially divided government was bending to Brussels' insistence on how negotiations will work.
Barnier said his priority was to clear up the uncertainties that the Brexit vote had created.
At their first meeting in Brussels on Monday, British and European Union negotiators agreed on a timetable for the Brexit talks. "David Davis said the row of the summer would be over the sequencing of Brexit talks, and one day in he has capitulated", said Tim Farron, outgoing leader of the Liberal Democrats.
David denied any retreat by his government, noting that "both sets of dialogue will continue once the European Union 27 will agree that divorce talks have reached "sufficient progress".
"No deal would be a very, very bad outcome for Britain, but there is a possible worse outcome and that is a deal that is deliberately structured to suck the lifeblood out of our economy over a period of time".
When asked whether agreeing to this schedule was a sign of the weakness of Britain's negotiating position, Davis said: "It is not how it starts but how it ends".
"The protection of the Good Friday agreement and the maintenance of the Common Travel Area are the most urgent issues to discuss", said Barnier, adding that "concrete and imaginative" solutions will be required to control goods and services without creating a hard border.
"Leaving gives us the opportunity to forge a bright new future for the United Kingdom - one where we are free to control our borders, pass our own laws and do what independent sovereign countries do", he said. "So my mind is not on making concessions or asking for concessions".
Theresa May's government already appears to be on the ropes with a failed attempt to strengthen her hand with a disastrous general election result and with further turmoil on the home front with the horrendous Grenfell Tower disaster.
A British source told Reuters that the mood on the first day of the talks was incredibly positive, with both the European Union and the United Kingdom recognising the need to move along with it quickly. Both the European Union and Britain said earlier this year that they want to keep it that way after Brexit.
'No doubt the road ahead will be challenging but, as Winston Churchill once said: a pessimist sees difficulty in every opportunity, an optimist sees opportunity in every difficulty'.
Mr Barnier quoted the founder of the trading bloc that later became the EU, Jean Monnet, as saying: "I am neither optimistic nor pessimistic".