The apology covers a practice that spanned the 1930s to the early 1980s, in which bereaved parents who lost babies in NHS hospitals were given no say in what would happen to their remains and no information about where they had been taken.
More than 1,000 stillborn babies were interred at three Wirral cemeteries, Frankby, Plymyard and Landican, between 1935 and 1981. A separate mass burial ground in Royton, Oldham, is believed to hold more than 270 babies.
Gina Jacobs, from Wirral, whose son Robert was stillborn in 1969, did not learn where her son was buried for 53 years. She has been among those campaigning for government recognition and received a personal apology from Cooper before the debate.
Cooper said: "On behalf of the government, I want to issue a profound apology to them." She added: "I have issued that apology to Gina today, but to so many more like Gina who have sought answers over many years, I'm sorry for the unthinkable pain caused by those historic practices, because what happened to those parents should never have happened and we need to ensure it can never happen again."
Jacobs told ITV News: "It didn't kind of sink in at first. I've wanted this, I've campaigned for this, I've begged for this, anyone who would listen, for four years now, and to actually have it said to me by the minister herself."
She added: "To me it's like a lottery win because I have wanted this. This brings dignity and respect to these babies, because they never got it at the time. And it brings it even to our missing babies."