Speaking on BBC Radio 4's Today programme, Mr Davis - who quit as Brexit secretary in protest at former prime minister Theresa May's Brexit plan - said it would be a "fair race" to reach a trade deal with the EU by the end of 2020 but "it can be done".
The UK is aiming to sign a permanent free trade agreement with the EU, along the lines of the one the EU has with Canada, by the end of the transition period in December.
Mr Davis said reaching a deal was "not a charitable exercise, this is an exercise of both sides recognising their own best interests".
European leaders have warned that the UK faces a tough battle to get a deal by that deadline.
Mairead McGuinness, the vice president of the European Parliament, said progress to agree a trade deal "might be left to the very last minute".
"Normally in trade negotiations we're trying to come together," she told BBC Breakfast. "For the first time we're going try and negotiate a trade agreement where somebody wants to pull away from us. I can't get my head around that and I think it's going to be quite complicated."
Celebrations and protests
Thousands gathered in Parliament Square to celebrate Brexit on Friday night, singing patriotic songs and cheering speeches from leading Brexiteers, including Nigel Farage.
The Brexit Party leader said: "This is the greatest moment in the modern history of our great nation."
Pro-EU demonstrators earlier staged a march in Whitehall to bid a "fond farewell" to the union.
Police in Whitehall arrested four men and also charged one man with criminal damage and being drunk and disorderly, while in Glasgow one man was arrested.
Meanwhile, other symbolic moments on a day of mixed emotions included:
The Union flag being removed from the European Union institutions in Brussels
The Cabinet meeting in Sunderland, the first city to declare in favour of Brexit when the 2016 results were announced
A light show illuminating 10 Downing Street and Union flags lining The Mall
A 50p coin to mark the occasion entering circulation
Meanwhile, US secretary of state Mike Pompeo said he was "pleased" the UK and EU had agreed a Brexit deal and the US would continue to build its "strong, productive, and prosperous relationship with the UK".
Washington's ambassador to the UK, Woody Johnson, said Brexit had been "long supported" by President Donald Trump.