The minister, Katherine Zappone, revealed the figure as she addressed the DÃ¡il (Irish parliament).
She made a statement following last week's revelation that "significant quantities" of human remains had been found at one of the homes in Tuam.
Ms Zappone said the Tuam discovery "confirmed what we had all feared".
The minister also paid tribute to the "tireless" County Galway historian Catherine Corless, whose personal research led to the inquiry and an excavation of site.
Ms Corless found death certificates for 796 infants who died at the Tuam home from a range of natural causes when she was writing a history of the site.
She continued to ask questions about what was done with their remains when she could not locate their burial records.
An inquiry, set up in 2014, confirmed last week that human remains had been found in "underground chambers" on the site of the demolished home.
The minister told the DÃ¡il that for former residents, campaigners, and Ms Corless it was a "moment of vindication".
"After decades, and years of hard work, determination and unwavering commitment, the truth has been laid bare for all of us to see.
"This House and our entire State owe a debt of gratitude to Catherine Corless for her work.
"Many men and women alive today spent time in that institution, either as children or as young women.
"Today I offer them my personal solidarity and, as a citizen, my personal apology for the wrongs that were done to them."
During the last century, thousands of pregnant unmarried women and girls were sent to mother-and-baby homes as it was deemed shameful to bear a child outside marriage.
Conditions in many of the institutions were harsh, and some former residents have said they felt "incarcerated" in the homes.
The home in Tuam, County Galway, was run by the Bon Secours order of nuns.
A child died there nearly every two weeks between the mid-1920s and 1960s.
Ms Zappone said it can "take time to shine a light on dark periods of our history" but added that what went on in the homes was "not without the support of many pillars in society".
"We must acknowledge that sometimes it was fathers and mothers, brothers and uncles, who condemned their daughters, sisters, nieces and cousins and their children to these institutions," the minister said.
"We must accept that between 1940 and 1965, a recorded 474 so-called "unclaimed infant remains" were transferred from mother-and-baby homes to medical schools in Irish universities."
She said the plight of residents of mother and babies homes was raised in the DÃ¡il in the 1950s.
"We must acknowledge that this very House debated legislation that allowed for those residing in institutions such as county homes to work for little or nothing in return for the so-called charity that was shown to them," Ms Zappone said.
"Lest we contend that people did not know what was happening, let us remember that some Members of this House spoke out against it."
The remains of 474 infants were transferred from mother-and-baby homes to medical schools over 25 years, the Irish minister for children has said.