Meanwhile, most MPs from the ruling Conservative Party voted against delaying Brexit - including seven cabinet members - meaning May had to rely on Labour and other opposition votes to get it through.
Brexit is only Brexit if it delivers full independence of all areas of government, law and public administration, her deal does not do that.
The prospect of a long extension period would be the stuff of nightmares for hardline eurosceptics, anxious that Brexit may never happen and could strengthen PM May's hand as she pushes to get her deal over the line.
The Council is due to meet next Thursday and Friday (21 and 22 March).
Thursday night, the British MPs also overwhelmingly rejected, by 334 votes to 85, an amendment presented by Sarah Wollaston, the Independent Group, proposing the organization of a second referendum on Brexit.
Hard Brexiteer Jacob Rees-Mogg tweeted: "A second referendum, the so called "losers" vote', has now been defeated in the House of Commons so is it is off the table".
May hopes the threat of a long delay will push Brexit supporters in her Conservative Party and members of the Democratic Unionists, the small Northern Irish party that props up her minority government in parliament, to back her deal at the third attempt.
The original government vote tonight was meant to give MPs a chance to delay Brexit - with some reports that Article 50 could be extended for as long as two years.
It revealed that leader Jeremy Corbyn and senior aides have met with backbenchers Peter Kyle and Phil Wilson, who are promoting a plan to accept Mrs May's deal on the condition that it is subject to a second referendum.
"Whilst much of the current impasse in Parliament is the fault of the Prime Minister who has refused to consider alternatives to her withdrawal agreement, I could not support a second referendum given the commitments I have made and from listening to the very many conversations, emails and letters that I have received from constituents. We don't want to leave without a deal but a lot will depend in terms of what the government can do on providing those guarantees that are necessary".
Mr Dodds spent Friday afternoon in meetings with key cabinet figures - including Chancellor Philip Hammond and Environment Secretary Michael Gove - as the government seeks to persuade MPs to support its deal when it returns to the Commons.
European Union leaders have said they would consider any request from London. That was rightly the focus of today's debate and votes and the government had been forced into extension after a succession of defeats. "In the last 24 hours Parliament has decisively rejected both her deal and no deal". That left Britain facing a disruptive "no-deal" exit on March 29, when a two-year countdown to departure runs out.
"Any Brexit deal looks less attractive when it is a concrete rather than abstract proposal".
The EU must increase pressure on Britain to move forward on its exit from the bloc, the European Parliament's Brexit coordinator told German television on Thursday, after the vote in the Commons.
"And we must all decide what we want our country to become".
In the event of a Brexit delay to end-2020, European Union leaders could instruct their officials to negotiate with Britain on their future relationship: "In these 21 months to the end of 2020, we could say that we really want to negotiate straight away on the future relationship".