Trade Secretary Liz Truss told the BBC the work had been "fully declared".
But she did not answer repeated questions on whether a Tory party donor initially provided the money to him.
It comes after the PM's former chief adviser, Dominic Cummings, claimed Mr Johnson had a "possibly illegal" plan for donors to pay the refurbishment.
Labour said questions remained, as details over how the work was paid for had yet to be made public.
The government is supposed to publish the list of minister's interests twice a year, but the last one showing money donated to them was released in July 2020.
Speaking on the BBC's Andrew Marr Show, Labour deputy leader Angela Rayner said Mr Johnson should come in front of MPs on Monday to answer questions and publish the list "as the public deserves [to see] it".
Mr Johnson lives in the flat above 11 Downing Street with his fiancee Carrie Symonds - a move echoing a number of his predecessors as it is bigger than the accommodation above No 10.
In March, Downing Street dismissed as "speculation" suggestions that refurbishments on the flat last year would be paid for by Tory donors through a charity set up for this purpose, amid reports that the costs had spiralled out of control.
But on Friday, Mr Cummings launched a blistering attack on his former boss via his blog, claiming he advised the PM the renovation plans were "unethical, foolish, possibly illegal and almost certainly broke the rules on proper disclosure of political donations if conducted in the way he intended".
A No 10 spokesperson said: "At all times, the government and ministers have acted in accordance with the appropriate codes of conduct and electoral law."
And Cabinet Office Minister Lord True told the House of Lords on Friday that "any costs of wider refurbishment in this year have been met by the prime minister personally".
Donations and loans to political parties of more than £7,500 must be reported to the Electoral Commission, which said it was having "discussions with the Conservative Party", but had not yet launched an investigation.
The controversy also comes amid an ongoing row over lobbying in Westminster, after revelations former prime minister David Cameron had texted ministers about the financial firm he was working for and businessman Sir James Dyson had directly contacted Mr Johnson over tax issues when working on ventilators during the pandemic.
By Damian Grammaticas, BBC political correspondent
The reason this issue is so difficult for Boris Johnson is it comes down to something simple - money.
How a job was paid for is a far more straightforward issue for people to form an opinion about than other recent questions about government conduct.
That swirl of revelation and allegation has dealt with the more intangible currency of politics, access, influence and favours.
This is about cash. Tens of thousands potentially.
The cost of a home makeover, and the as-yet-unanswered question, did Mr Johnson get help?
And if he did, what about the implications for how that benefit should be declared - whether for electoral or tax reasons.
Also talking to Marr, Ms Truss defended Mr Johnson, saying: "My understanding is the costs have been covered by the prime minister and everything has been fully declared by the rules."
But asked five times whether the money was put up by a party donor first, she would not answer.
Instead, she told the programme: "What I know is the prime minister has personally met the costs of the flat refurbishment and that is what people in Britain want to know and that is what has happened."
When asked why the list of minister's interests had yet to be published, Ms Truss replied: "I'm sure it will be published in line with the rules."
And questioned about rumours of leaks out of No 10 - also addressed by Mr Cummings in his blog - the minister urged Mr Johnson to "carry on" doing "an excellent job" rather than "listening to these noises off".
She added: "I think this a complete load of Westminster tittle tattle that people simply don't care about."
But Ms Rayner said there was a "real stench" around the government and the "serious allegations" it faces.
Ms Rayner told Marr: "The issues that we have seen with ministers over the last couple of months has really undermined the institution of government and there are serious questions now the Conservatives need to answer.
"These are serious allegations and we think the prime minister should come to the House of Commons on Monday and should declare the ministers register of interests, which is eight months in delay.
"The problem is the government is hiding."
The SNP's Westminster leader, Ian Blackford, backed the call for the prime minister to answer questions in Parliament.
He said: "The longer Boris Johnson stays in hiding and dodges accountability, the more overpowering the stench of sleaze surrounding him and his government becomes.
"Rather than hiding behind Tory ministers, Boris Johnson must now come before Parliament and answer these very serious questions for himself."