Sajid Javid has apologised to 18 members of the Windrush generation after a review found they may have been wrongfully removed or detained.
The home secretary said what happened was “completely unacceptable”.
Labour's Shadow Home Secretary Diane Abbott said the apology was "overdue and is nowhere near good enough".
In a statement issued on Tuesday, the Home Office said 11 of the 18 people identified had voluntarily left the UK after wrongly being told they could not stay.
The department said the remaining seven were detained but subsequently released without being removed.
The migrants came to the UK from the Caribbean before 1973 but were unable to demonstrate their continuous residence here.
Of the 18, four were removed and two were detained before May 2010 and seven were removed and five detained after May 2.
So far the Home Office has contacted 14 out of the 18.
Javid said: “I would like to personally apologise to those identified in our review and am committed to providing them with the support and compensation they deserve.
“We must do everything we can to ensure that nothing like this happens again – which is why I have asked an independent adviser to look at what lessons we can learn from Windrush.”
He added: “The experiences faced by some members of the Windrush generation are completely unacceptable and I am committed to righting the wrongs of the past.”
The Windrush scandal which hit the government ultimately led to the resignation of Amber Rudd as home secretary.