Those infected were treated with blood-clotting agents, a Downing Street spokesman said Tuesday.
Some 2,400 people died as a result of the tainted blood treatment, "and it was necessary to establish the causes of this appalling justice," the spokesman said.
Those affected will help decide what form the inquiry will take, he said. The inquiry will span the entire United Kingdom.
The announcement was made during a Cabinet meeting by Prime Minister Theresa May in consultation with Health Secretary Jeremy Hunt, the spokesman said, and came hours before the UK Parliament was due to hold an emergency debate on whether there should be such an inquiry.
The infections took place in the late 1970s and 1980s, when more than 4,500 people with hemophilia and other bleeding disorders were infected with HIV, hepatitis B and C, and a range of other blood-borne viruses, according to the UK's Haemophilia Society
Of the 1,200 people infected with HIV during this period, fewer than 250 are still alive.
People suffering from bleeding disorders often lack proteins in their blood that help their blood to clot, meaning even the smallest of injuries can lead to excessive bleeding. Treatment involves transfusion of these proteins into a patient's blood from donors, such as a protein known as factor 8 for treatment of hemophilia A.
In the 1970s, these new treatments were produced by pooling human blood from as many as 40,000 donors and concentrating it to extract the required "factor," or protein, according to the Haemophilia Society.
"Blood and blood products were known to transfer viruses such as hepatitis and these risks was vastly increased when they were pooled using the new techniques," the Haemophilia Society said in a statement.
When supplies of UK-produced factor concentrates were not sufficient to meet demand, health authorities from the UK's National Health Service began to import from the US.
"In the US, high-risk paid donors were used as well as being collected in prisons increasing the risk of contamination with blood-borne viruses," the Haemophilia Society said.
The need to investigate
Still, there is a need to examine UK producers, said Scottish Nationalist Party health spokeswoman Dr. Philippa Whitford, who also is a surgeon.
"The UK producers have often been found wanting in the quality of product that they came up with. So we mustn't pat ourselves on the back and imagine that the UK product was somehow safe and this was all due to the US," she told the Press Association.
No government, health or pharmaceutical company has ever been prosecuted or admitted liability for the scandal in the UK.
"It is a tragedy that has caused unimaginable hardship and pain for all those affected," the Downing Street spokesman said. "A full inquiry to establish the truth behind what happened is the right course to take."
"The Prime Minister has earned a place in history as someone who listened on an issue that many had ignored, and put party politics aside in the cause of giving people their basic right to answers," Diana Johnson, a Labor MP who has campaigned on the issue, said in a tweet.
Survivors and relatives of those who died should be part of the inquiry which should investigate "not just the lead-up to this tragedy, but the aftermath -- including the alleged criminal cover-up; and the loss of documents and medical records," the lawmaker said.
Andy Burnham, a former MP who is now the mayor of Manchester, had threatened criminal action unless action was taken before the summer recess.