"There is more that unites us than divides us", he said, adding that Britain was looking for a "positive and constructive tone" in the talks.
Yet many in Brussels fear that London has no real strategy, with May under pressure at home, still trying to close a deal with a conservative Northern Ireland party to stay in power, and facing criticism for her handling of the aftermath of a devastating tower block fire.
European Union negotiator Michel Barnier said the negotiations which should lead to a breakup by March 2019 "must first tackle the uncertainties caused by Brexit - first for citizens, but also for the beneficiaries of the European Union policies and for the impact on borders, in particular Ireland".
Both sides say they want to safeguard the rights of millions of citizens who have settled in Britain or Europe.
"The result of the election seems to be a vote against "no deal is better than a bad deal" and against a hard Brexit".
Significantly, Mr Davis said he would tell Labour and other parties about the Government's plans in advance - reflecting its weak position, with no Commons majority.
The negotiations start against the backdrop of a mounting domestic crisis in the U.K. May's weakened post-election standing deteriorated further last week after she was slammed for misjudging the national mood following a deadly tower block inferno in West London.
Arriving Monday morning at the Berlaymont, Barnier offered his condolences to the United Kingdom over the latest terror attack, and to Portugal over the fatalities in recent forest fires.
Unresolved is everything from the status of European Union citizens living in Britain, to intelligence sharing, to the future of tens of thousands of British jobs that could be wiped out if businesses move to Europe to avoid new trade barriers.
How the negotiations progress will be monitored closely by investors and business leaders. The pound has fallen about 14 percent against the dollar since the June 23 referendum, pushing up inflation even as the economy shows signs of slowing. It is a member of the EU's customs union, meaning Turkish exports to the EU are tariff-free.
Britain's Secretary of State for Exiting the European Union David Davis speaks after the first day of Brexit talks in Brussels.
"We have to commit ourselves now mutually to guarantee rights to citizens on either side of the Channel so they can continue their lives as in the past", Mr Barnier said.
"That has always been our first aim and that is what we will do".
A key issue he did not mention was the EU's bill for Britain to leave, which Brussels estimates at a colossal 100 billion euros.
Talks in Spain are expected to include the thorny issue of Gibraltar after the EU's guidelines for talks suggested the disputed territory's future economic status might be subject to a veto from Madrid. As in any divorce, count on both sides to be picky in splitting the goods and dues.