Mr Davis said that while there was a long way to go - a conclusion is due by March 2019 - the negotiating teams had got off to a "promising start; we have taken the first, critical steps together".
On migration, Mr Hammond said it will be managed but not "shut down".
What's more, Davis made clear that there's little appetite in Britain - at least in the opening salvo of negotiations - for a so-called soft Brexit. Rather than sovereignty or controlling immigration, which are the issues likely to motivate other colleagues in the Cabinet and certainly in the Conservative Party. Agreeing to pay a "Brexit bill" may be more inflammatory.
In the recent election the two main parties shared 82% of the vote, with both affirming the intention to leave, and to leave the single market and customs union as well.More news: Fed: Biggest US banks strong enough to withstand recession
He also signalled the United Kingdom would try to maintain the "frictionless" border arrangements of the EU's customs union for an "implementation period" after leaving.
First, he hoped to secure a "comprehensive trade agreement for trade in goods and services".
"The collective sigh of relief would be audible. The benefit to our economy would be huge", he claimed. "But ensuring that it happens via a smooth pathway to a deep and special future partnership with our European Union neighbours, will require every ounce of skill and diplomacy that we can muster".
Mr Hammond emphasised that leaving the EU meant the country would leave the single market and the customs union.
Officials on both sides play down expectations for what can be achieved in one day.More news: President Trump Admits: 'I Am Being Investigated for Firing the FBI Director'
"To do this in the context of our wider objectives will be challenging". This involves establishing new regulatory guidelines for businesses; conducting risk management; and reassuring businesses that the new framework will be permanent. EU leaders want May to lay off threats that she would walk out and leave a chaotic legal limbo for all Europeans.
"I thought we had won that argument".
The Merkel ally felt confident about the bloc's position, claiming that the new-found stability gained through positive electoral results in France and the strengthening positions of key members such as Germany, Italy and Spain will help in contrast to the chaos taking over London in the aftermath of the snap general elections.
While Barnier insists on the "sequencing" of talks, so that trade negotiations can not start until probably January, finding a way to avoid a "hard" customs border for troubled Northern Ireland may well involve some earlier discussion of the matter. The insistence of European Union figures and government to agree the "divorce terms" first, means agreeing a full trade deal by March 2019 is now highly unlikely.More news: Qatar won't cut gas to UAE: Qatar Petroleum CEO
"We are citizens first".