In a joint press conference with EU negotiator Michel Barnier, the Brexit Secretary said there had been progress on citizens’ rights, although difference remained.
Mr Davis said EU citizens who have permanent residency documents will not have to reapply and others could invoke their rights at UK courts.
The Brexit Secretary said there were now the foundations for a deal to “end the anxiety” of EU citizens in the UK.
And he said discussions on a financial settlement had also been “very constructive” while there was scope to “explore” a transition deal.
Mr Davis said: “We have made important progress and capitalised on the momentum created by the Prime Minister’s speech.
“We are working quickly through a number of complex issues but there remain some points where further discussion and pragmatism will be required to reach an agreement.”
The pair appeared in Brussels on the final day of the latest round of talks, in a much more friendly exchange than the frosty scenes last month.
But both men admitted there were gaps between them on future citizens’ rights, which the EU is insisting must be enforced through the European Court of Justice.
Mr Barnier said there was clarity on certain points but work remains to be done on talks.
He said: “The Prime Minister’s speech in Florence has created a new dynamic in the negotiations.
“We have managed to create clarity on some points but for others we are not there yet and more work remains to be done.
“We will keep working with a constructive spirit until we reach a deal on the essential principles on the UK’s withdrawal.”
He said: “On the financial settlement, experts have had detailed talks on the aspects and those talks were useful.
“Prime Minister May said two things in Florence; no member state should pay more and no member state should receive less because of Brexit.
“The second – the UK will honour commitments taken during its membership.
“For the EU the only way to reach progress is for all commitments to be honoured.”
Mr Barnier said they had had a “constructive” week, but they were not yet there with regards to achieving sufficient progress with talks.
He said: “More work is needed in the coming weeks and the coming months.
“I hope the new dynamic created by Theresa May’s speech will continue to impact our work.”
During their last tense press conference together on August 31, Mr Davis demanded that the EU be more “imaginative” and flexible” over the negotiations.
But his Brussels counterpart hit out at what he claimed was no “decisive progress” in the discussions.
When the fourth round of talks began on Monday, a bullish Mr Davis claimed there were “no excuses” for standing in the way of progress after Prime Minister Theresa May’s speech in Florence.
But Mr Barnier struck a more cautious tone, saying real progress on the divorce bill, EU citizens’ rights and the Irish border was “essential” before the discussions could move on to trade.
Mrs May’s Florence speech has been welcomed as big step forward in the Brexit process.
The Prime Minister outlined a number of commitments, including continuing to pay in to the EU budget after 2019 as part of a two-year transitional arrangement.
But, despite the apparent progress, Brussels chiefs have warned that there is no guarantee the Uk will successfully broker a transition period.
On Tuesday, the Prime Minister met EU Council President Donald Tusk at Downing Street in the hope of moving the process forward.
Mr Tusk claimed the UK had given up the idea of “having its cake and eating it” following the Florence speech and its approach had become more realistic.
At the press conference last month, a visibly frustrated Mr Barnier hit out at a lack of movement on the EU’s red line issues.
He told the audience: “Over the course of this week we have made a number of useful clarification on a number of points, for instance the status of border workers.
“However, we did not get any decisive progress on any of the principle subjects, even though on the discussion we had about Ireland – that discussion was fruitful.”