European Council president Donald Tusk made the announcement on Monday morning which means Britain will remain in the European Union until next year - unless Parliament ratifies Boris Johnson's deal sooner.
The Prime Minister has previously said he would prefer to be "dead in a ditch" than miss the October 31 deadline.
Mr Tusk tweeted: "The EU27 has agreed that it will accept the UK's request for a #Brexit flextension until 31 January 2020.
"The decision is expected to be formalised through a written procedure."
In response, Mr Drakeford called for a People's vote saying, "We have seen three wasted years of delay and bluster from UK Government – it must finally be time to take the decision back to the people and remain must be on the ballot paper."
Following the announcement, the Welsh Conservatives said "People don't want more delays" and called for an election to take place.
"They want to get Brexit done so that we can focus on investing in our public services and continue to hold this failing Welsh Labour Government to account.The only solution to this Brexit impasse is for a general election."
Plaid Cymru’s Liz Saville Roberts MP also said opposition parties should use the extension to work together to deliver 'People’s Vote' which seeks to ensure any deal is put before the country in a public vote.
She said, “Boris Johnson has failed to deliver on yet another promise. Working together, opposition parties have stopped Mr Johnson delivering his damaging Brexit deal – or no deal – on Halloween.
“Those of us opposed to inflicting the harm of Brexit on the four nations of the UK must now unite again to deliver the most sensible way to end this mess – a final say referendum."
Leader of the Welsh Liberal Democrats Jane Dodds said the "lifeline" given to the UK by the EU must be used to "break the Brexit logjam and take no deal off the table once and for all".