The UK stopped following EU rules at 23:00GMT (midday NZ time), as replacement arrangements for travel, trade, immigration and security co-operation came into force.
Boris Johnson said the UK had "freedom in our hands" and the ability to do things "differently and better" now the long Brexit process was over.
French President Emmanuel Macron said the UK remained a "friend and ally".
UK ministers have warned there will be some disruption in the coming days and weeks, as new rules bed in and British firms trading with the continent come to terms with the changes.
Officials have insisted new border systems are "ready to go" amid fears of hold-ups at ports.
The UK officially left the 27-member political and economic bloc on 31 January, three and half years after the UK public voted to leave in the 2016 Brexit referendum.
But it has stuck to the EU's trading rules for the past 11 months while the two sides negotiated their future economic partnership.
After trade talks went down to the wire, a landmark treaty was finally agreed on Christmas Eve. It became law in the UK on Wednesday after it was approved by Parliament.
Under the new arrangements, UK manufacturers will have tariff-free access to the EU's internal market, meaning there will be no import taxes on goods crossing between Britain and the continent.
But it does mean more paperwork for businesses and people travelling to EU countries.
PM hails 'amazing moment'
Johnson - who was a key figure in the Leave campaign during the 2016 referendum and who took the UK out of the EU in January six months after becoming prime minister - said it was an "amazing moment" for the UK.
In his New Year message, the prime minister said the UK was now "free to do things differently, and if necessary better, than our friends in the EU".
"We have our freedom in our hands and it is up to us to make the most of it," he said.
Veteran Conservative MP Sir Bill Cash said the culmination of the Brexit process was a "victory for sovereignty and democracy".
Opponents say the country will still be worse off than it was while in the EU - and there is still uncertainty about what it will happen to banking and services, which are a major part of the UK economy.