But Foreign Secretary Dominic Raab said he was confident enough MPs would back the deal next week, and Brexit would still happen by the deadline.
Labour, however, has said it will back moves to put the deal to a referendum.
No 10 said the PM sent "Parliament's letter" to Brussels - unsigned - and accompanied it with a second letter - which was signed - explaining why he believed a delay would be a mistake.
The government has vowed to press ahead with the legislation - the Withdrawal Agreement Bill (WAB) - to implement the Brexit deal next week.
Why has the PM asked for another extension?
Having reached a new Brexit deal with the EU last week, the prime minster had intended to bring it to Parliament and ask MPs to approve it in a so-called "meaningful vote".
However, in the first Saturday sitting in the Commons for 37 years, MPs instead voted in favour of an amendment withholding approval of the deal until all the necessary legislation to implement it had been passed.
Tabled by Tory MP Sir Oliver Letwin, the amendment was intended to ensure that Mr Johnson would comply with the terms of the so-called Benn Act designed to eliminate any possibility of a no-deal exit on 31 October.
Under that act, Mr Johnson had until 23:00 BST on Saturday to send a letter requesting a delay to the UK's departure - something he did, albeit without his signature.
Mr Raab told the BBC's Andrew Marr show that "notwithstanding the parliamentary shenanigans, we appear to have now the numbers to get this through".
He said there were "many people in the EU" who were "deeply uncomfortable" about a further delay to Brexit and urged MPs to "get on, get it through the House of Commons, and move on."
His colleague, Michael Gove, chancellor of the Duchy of Lancaster, accused those who backed the Letwin amendment of voting "explicitly to try to frustrate this process and to drag it out".
He told Sky News' Sophy Ridge on Sunday "we have a deal that allows us to leave" on 31 October, but the government would now trigger Operation Yellowhammer - its contingency plan to handle a no-deal Brexit - because there was no guarantee the EU would grant an extension.
However, it will be up to Commons Speaker John Bercow to decide whether to allow that vote.
What is Labour saying?
Labour had planned to vote against Boris Johnson's deal - although a few rebels would likely have backed it - arguing it would be bad for the economy, jobs, workers' rights and other areas like the environment.
Shadow Brexit secretary Sir Keir Starmer accused the prime minister of "being childlike" by sending a second letter to the EU contradicting the first stipulated by the Benn Act.
He told Andrew Marr his party would seek to amend the deal when the WAB is brought to Parliament, for example by demanding a UK-wide customs union with the EU and single market alignment.