In the wake of the defeat, opposition parties called on the Government to take a no-deal Brexit off the table in a bid to work cross-party on a way forward but Mrs Leadsom said that was “not possible”.
The Cabinet minister told MPs that removing no deal as an option would be “an incompetent thing to do”.
Mrs Leadsom, speaking at Business questions in the Commons, said: “Having passed the EU (Withdrawal) Act, it is the case that the legal default is that the UK will leave the European Union on March 29, and if a deal has not been voted for then it will be with no deal unless other alternative arrangements are put in place.
“The people have spoken and it is our job to fulfil in line with the requirements of the people, this House is a servant of the people of this country, the entire United Kingdom.”
She later added: “If you take no deal off the table and you stop preparing for no deal, then for a sensible Government that would be totally an incompetent thing to do.
“Government has to continue to prepare for all eventualities including no deal, it is not possible to remove no deal from the table and still abide by the will of the people as expressed in the referendum.”
The rebuttal came after shadow Commons leader Valerie Vaz asked Mrs Leadsom if the Government would consider extending Article 50.
Ms Vaz used Business questions to highlight the Government’s record-breaking run, and said: “This is the first Government to be held in contempt of Parliament, the Prime Minister had a vote of no confidence from within her own party, a vote of no confidence in the Government which they won yesterday because they have a confidence and supply agreement, but, yet again breaking records, 432 MPs voted against the Prime Minister’s deal - the biggest defeat of a government in history.”
SNP Commons leader Pete Wishart also mocked ministers, who he said were treating the Brexit defeat as a “mere flesh wound” - “Like Monty Python’s Black Knight - armless and legless and fights on prepared to bite the nation into submission.”
Mrs Leadsom earlier told MPs they will debate the Government’s proposed Brexit “next steps” on January 29.
She said: “A statement and a motion on the Government’s next steps under Section 13 of the EU (Withdrawal) Act will be tabled on Monday.
“A full day’s debate on the motion will take place on Tuesday January 29, subject to the agreement of the House.”
Mrs Leadsom went on to quote from Winnie-the-Pooh and also wished happy birthday to Speaker John Bercow.
The pair have clashed several times of late in the Commons.
She said: “Yesterday is history, tomorrow is a mystery and today is a gift, which is why we call it the present.
“On the eve of AA Milne’s birthday that’s a favourite quote from Winnie-the-Pooh and, as Eeyore said, ‘it never hurts to keep looking for sunshine’ and so can I wish you, Mr Speaker, a very happy birthday for Saturday.
“Finally, can I leave the House with an uplifting and rather wise thought from Winnie-the-Pooh - ‘If the person you are talking to doesn’t appear to be listening, be patient, it may simply be he has a small piece of fluff in his ear’.”
Shadow Defra minister David Drew (Stroud) raised fears that the Commons would run out of time to get the necessary legislation through before the UK is due to leave the EU.
He said: “Before March 29, Defra alone has something like 80 statutory instruments to get through and we’ve also got the Agriculture Bill and Fisheries Bill to come back on report and third reading.
“The Leader of the House might be Superwoman but how is she going to do that?”
Mrs Leadsom replied that she was “absolutely not” Superwoman but did review the timetable daily and was satisfied there was enough time.
“Obviously we are not flush with time,” she said. “I do believe, however, we have enough time to get all of our secondary legislation through, and indeed the primary legislation that needs Royal Assent, by departure date.
“I continue to keep that under review and it will require the cooperation and support of the House.”