The bill has set up a redress scheme, under which any mother or child who spent time in an institution will be entitled to a standard payment of £12000.
£2000 will be paid to the family members of mothers and children who have died since 28 April 1953.
The devolved government estimates the scheme will receive around 10,000 applications, with payments totalling £90m.
It is understood the public inquiry is expected to last around three years and will cost around £14m.
The inquiry will further investigate the issues raised in a report due to be published soon, by the Truth Recovery Independent Panel.
It was commissioned to gather evidence in a non-confrontational setting.
The work to design the investigatory process began after research by university academics in 2021, external.
It found that more than ten thousand women and girls spent time in the institutions, and that a number of them had become pregnant as the result of sexual crime.
First Minister Michelle O’Neill said the institutions were “built on the foundations of systemic misogyny”.
“Within their walls, women and girls were stripped of dignity, silenced and shamed.
“This legislation is another step towards putting that right through truth, acknowledgement and redress.”
Deputy First Minister Emma Little-Pengelly said: “While legislation alone cannot undo the suffering endured across all parts of our society, we hope this marks the beginning of a new phase.”
The Salvation Army said: “We ran mother and baby homes with the intention of caring for vulnerable women and children but testimonies like these show that there were people in our care whom we let down and for that we are profoundly sorry.
“We are deeply saddened to hear of the traumatic experiences that some people endured in the care of The Salvation Army many decades ago; the hurt of which they still carry.
“We welcome the various processes and inquiries underway to understand more about how women and their children were treated in Mother and Baby Homes.
“It is right that we should be transparent about the times in our past when we failed to provide the support and care that people needed and deserved.”
The Good Shepherd Sisters have previously said they would offer the investigation their “fullest cooperation” and they regretted they “could not and did not always meet the multi-faceted needs of these women”.
BBC News