The Human Tissue Act prioritises the wishes of deceased patients for the first time, meaning that medics can use their organs if they have said they want to donate, even if the family objects.
Relatives currently block the removal of organs from the corpses of registered donors in an estimated 10 percent of cases.
The law change is expected to ease the chronic shortage of donor organs in Britain, where about 500 people die every year waiting for a match.
It will also allow kidney and other live donations between total strangers with matching tissue types.
Currently, such transplants can only be made between genetic relatives and people with proven close relationships such as spouses.
The law change came about following reviews into the organ retention scandal at Alder Hey Children's Hospital in Liverpool, northwest England.
Hearts and other organs were harvested from the bodies of babies who died at the hospital between 1988 and 1996, without the knowledge or consent of their families.
In a further change, medical schools, museums and other organisations which store organs and body parts will have to be licensed.
Exhibitions such as German anatomist Gunther von Hagens's controversial Body Worlds, which displays preserved human corpses in action poses, will also have to meet these requirements.
The law, which applies to England, Wales and Northern Ireland, also makes collecting DNA samples without permission illegal, with some exceptions including police inquiries.
Separate legislation is already in place in Scotland, where health is controlled by the Scottish Executive in Edinburgh.