He said she has taken “significant steps” to address the problem of overcrowding at the Manston migrant processing facility in Kent.
Listing her actions, he told MPs at Prime Minister’s Questions: “Since September 30, more hotels with 4,500 new beds, appointing a senior general to control the situation at Manston and, indeed, increasing the number of staff there by almost a half.
“These are significant steps that demonstrate that we are getting a grip of this system.
“But this is a serious and escalating problem. We will make sure that we control our borders and we will always do it fairly and compassionately, because that is the right thing.”
Mr Sunak was challenged at PMQs by Sir Keir Starmer, who said responsibility for an asylum system which Ms Braverman described as “broken” lies with the Tories, who have been in power since 2010.
The Labour leader said just 4% of the asylum claims made by people who crossed the English Channel in small boats last year have been processed.
He added: “They’re only taking half the number of asylum decisions that they used to. That’s why the system is broken.”
Sir Keir asked whether Ms Braverman had been given legal advice to move people out of the overcrowded Manston facility,
The Home Secretary has insisted she has not ignored advice about the requirement to relocate detainees, but Mr Sunak would not comment on the legal advice.
The Labour leader shot back: “I think the answer to the question whether the Home Secretary received legal advice to move people out of Manston is yes – he just hasn’t got the guts to say it.”
Earlier, a Cabinet minister acknowledged that both the French and UK authorities need to do more to tackle the problem of migrants risking their lives to cross the English Channel in small boats.
Around 40,000 migrants have crossed the Channel so far this year, with Ms Braverman warning of an “invasion” on the south coast, comments which have been condemned by opponents.
Provisional Government figures to date show 39,913 people have arrived in the UK after making the journey.
No crossings were recorded by the Ministry of Defence (MoD) on Tuesday and 46 people arrived in one boat on Monday.
In October alone 6,912 people made the journey, with 1,065 arriving in a single day. This is the third highest monthly total this year, after 8,641 were recorded in August and 7,961 were recorded in September, PA news agency analysis of the figures show.
Transport Secretary Mark Harper told Sky News: “I think both countries, Britain and France, could do more. What we need to do is work with the French, they do a lot already.
“We provide resources to help them and, of course, people will know our border controls in France are actually physically located in France, and we’ve always worked in close partnership with French authorities.
“Do we think they could do more? Yes. We could do more as well. It’s about improving that partnership.”
Military personnel were involved in efforts on Tuesday to help move migrants away from Manston.
Immigration minister Robert Jenrick said the numbers at the site had “fallen substantially” on Tuesday, with more expected to be moved on Wednesday.
Sir Roger Gale, Tory MP for North Thanet, which includes the site, welcomed the development, having clashed with Ms Braverman about her handling of the chaos at Manston.
He praised Mr Jenrick for “rectifying the mistakes that have been made by others”, adding “this must never be allowed to happen again”.
It is unclear whether those moved from Manston will be taken to hotels or alternative accommodation.
The Government’s efforts to secure hotel space have also faced challenges in the courts.
Sheila Oxtoby, chief executive of Great Yarmouth Borough Council, which won a legal case against the Home Office, said the local authority objected to having decisions about the use of hotels in tourist areas “imposed” on it.
She told BBC Radio 4’s Today programme: “We have always, right from the beginning, offered to work with the Home Office to find the most suitable accommodation and the best solution for both the asylum seekers and the existing community – but that has largely fallen on deaf ears and our offer has not been taken up.”