Leaders of the 27 other members of the EU took less than 40 minutes to approve the draft documents at a special summit in Brussels, EU Council president Donald Tusk confirmed.
Theresa May, who met with Mr Tusk and the presidents of the European Parliament and Commission on Saturday evening, was not present but will join the summit later on Sunday morning.
The decision in Brussels clears the way for the deal to be put to parliamentarians in Brussels and at Westminster, where the Prime Minister faces a difficult job getting it past her own MPs.
Mrs May issued a direct plea to the public to support her Brexit deal, saying it would mark "a new chapter in our national life" and bring about a moment of "renewal and reconciliation" in a letter to the nation.
Eurosceptic Tory MP Andrew Bridgen, a prominent critic of the Prime Minister, said: "With regard to the Withdrawal Agreement and the proposed future relationship 'never in the history of our great nation have so many been betrayed, so deliberately, for so little'."
But Mrs May said the deal "will honour the result of the referendum" by allowing the UK to "take back control" of its money, laws and borders.
"It is a deal for a brighter future, which enables us to seize the opportunities that lie ahead," she said.
She promised she would be "campaigning with my heart and soul" to win the vote in the Commons.
And "with Brexit settled" the UK will be able to focus on the economy, NHS, building homes and tackling the "burning injustices" in society, the Prime Minister added.
The summit in Brussels is only taking place after Spain claimed the UK and EU had given into its demands for concessions over the future of Gibraltar.
Spanish Prime Minister Pedro Sanchez claimed the UK and European Union had agreed to its demands for guarantees over the status of Gibraltar in future negotiations.
Madrid's foreign minister Josep Borrell went further, saying the agreement is "highly positive for Spain" and "the most important" since the Treaty of Utrecht in 1713 under which Gibraltar was ceded to the UK.
But Mrs May told reporters in Brussels: "The UK's position on the sovereignty of Gibraltar has not changed and will not change.
"I am proud that Gibraltar is British and I will always stand by Gibraltar."
And Gibraltar's Chief Minister Fabian Picardo used a live TV address to dismiss Mr Sanchez's claims: "What you have heard from the Spanish Prime Minister today was not a reflection of any new position, however much he tried to present it as such."
The developments came ahead of elections in Andalusia, southern Spain, in December with speculation that Mr Sanchez was trying to use the situation for internal political reasons.
In an effort to resolve the row a letter from the UK's ambassador to the EU Sir Tim Barrow confirmed the Withdrawal Agreement imposes no obligations regarding the "territorial scope" of future agreements - Spain wants to make sure it has a veto over how any UK-EU trade deal applies to the Rock.
But the UK has also made clear it will negotiate future agreements on behalf of all territories for whose external relations it is responsible - including Gibraltar.
Tory Eurosceptic Nadine Dorries said: "May has capitulated to every single EU demand. She's abandoned NI, and now Gibraltar.
Liberal Democrat Brexit spokesman Tom Brake claimed the Prime Minister had "caved in" and "appears to have cast the people of Gibraltar aside".
Mrs May's political difficulties at home show no sign of abating, with speculation that Remain-leaning Cabinet ministers have begun secret talks on a Brexit "Plan B" if the deal is rejected by MPs.
The Sunday Times reported that Chancellor Philip Hammond - who warned on Saturday of "economic chaos" if there is no deal - is working with David Lidington, Amber Rudd, Greg Clark and David Gauke on trying to persuade Mrs May to adopt a softer Brexit which may be able to get through Parliament if her plan is rejected.
The Sunday Telegraph also reported that "several senior ministers" were working on plans for a Norway-style relationship with the EU.