He was appointed after Home Secretary Amber Rudd's late night resignation after she said she had "inadvertently misled MPs" over what she knew about immigration removal targets which she had earlier denied existed.
Javid said the interior ministry is investigating about 2,500 cases and has resolved about 100 cases.
"I will do whatever it takes to put it right".
May replied that she believed Rudd had given her evidence "in good faith"; but that she could "understand why, now you have had chance to review the advice that you have received on this issue, you have made the decision you have made, and taken responsibility for inadvertently misleading the home affairs select committee".
She felt it "necessary" to tender her resignation after the emergence of documents showing those goals were in place. As the Home Office didn't record details of each individual, those who didn't get documents are having a hard time to prove they are here legally.
Named after the British ship Empire Windrush, which arrived at Tilbury Docks in Essex with 492 Caribbean passengers in 1948, the Windrush immigrants never formalised their status, often because they were children who came over on their parents' passports and then never applied for their own.
Javid, whose family came to the United Kingdom in the 1960s, is now Communities, Local Government and Housing Secretary, reports the BBC.More news: PM Modi: Female athletes' success made Commonwealth Games 2018 'special' for India
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Javid, a lukewarm campaigner to stay in the bloc, replaces Amber Rudd, one of the cabinet's most outspoken pro-European members.
Mr Javid, the first lawmaker from Britain's black, Asian and minority ethnic community to become home secretary, tried before his appointment to defuse public anger over the scandal by saying his own family could have been caught up in it.
Born in Rochdale and raised in Bristol, Mr Javid attended a local state school before he went on to read Economics and Politics at Exeter University.
Explaining the thinking behind Mr Javid's appointment, Mrs May's spokesman said he was "one of the most experienced ministers" in Cabinet who had "proved his drive, his ambition and his determination to get to grips with hard subjects". My first priority is to make sure the Home Office always does all it can to keep British people safe.
"I had numerous dealings with him over the local government finance settlement and he came across as a very hard negotiator, but also very fair and effective".
"Just as importantly, people who have come here legally and enriched the life of our country should not expect the state unreasonably to challenge their presence here; rather, it should help them prove their right to continue living here and contributing to the life of our nation".
He told MPs: "I want to start by making a pledge, a pledge to those from the Windrush generation who have been in this country for decades and yet have struggled to navigate through the immigration system". In doing that I am really privileged to have a fantastic group of people here, the staff here, who together will work to make our country even stronger.