Mr Hunt told the Commons that a "computer algorithm failure" dating back to 2009 had meant many women aged 68 to 71 in England were not invited to their final routine screening.
It is not known whether any delay in diagnosis resulted in avoidable death, but it is estimated that between 135 and 270 women had their lives shortened as a result, he said.
An independent review has been launched into the "serious failure" in the programme, run by Public Health England (PHE).
Mr Hunt said "administrative incompetence" meant some families may have lost, or be about to lose, a loved one to cancer.
Women in England between the ages of 50 and 70 are currently automatically invited for breast cancer screening every three years.
The issue was first brought to the attention of the Department of Health and Social Care in January, but was initially thought to pose a "limited" risk to patients.
It was escalated to ministers in March by PHE following an urgent clinical review, with the Government told the error should not be made public to ensure existing screening services were not overwhelmed.
Mr Hunt told the Commons: "Earlier this year PHE analysis of trial data from the service found that there was a computer algorithm failure dating back to 2009.
"The latest estimates I have received from PHE is that as a result of this, between 2009 and the start of 2018 an estimated 450,000 women aged between 68 and 71 were not invited to their final breast screening.
"At this stage it is not clear whether any delay in diagnosis resulted in any avoidable harm or death and that is one of the reasons I am ordering an independent review to establish the clinical impact."
It is currently estimated that between 135 and 270 women may have had "their lives shortened as a result".
He added: "I am advised that it is unlikely to be more than this range and may be considerably less.
"However, tragically there are likely to be some people in this group who would have been alive today if the failure had not happened."
Of those who missed invitations, 309,000 are estimated to still be alive and all those living in the UK who are registered with a GP will be contacted before the end of May.
All women who were not sent an invitation for their final screening will be given the opportunity to have a new appointment.
Those under the age of 72 will receive an appointment letter informing them of the time and date, while those over 72 will also be offered a screening and have access to a helpline to decide if it will be beneficial.
Mr Hunt said: "Irrespective of when the incident started, the fact is that for many years oversight of our screening programme has not been good enough.
"Many families will be deeply disturbed by these revelations, not least because there will be some people who receive a letter having had a recent diagnosis of breast cancer.
"We must also recognise that there may be some who receive a letter having had a recent terminal diagnosis.
"For them and others it is incredibly upsetting to know that you did not receive an invitation for screening at the correct time and totally devastating to hear you may have lost or be about to lose a loved one because of administrative incompetence.
"So on behalf of the Government, Public Health England and the NHS, I apologise wholeheartedly and unreservedly for the suffering caused."
The review, expected to report in six months, will be chaired by Lynda Thomas, the chief executive of Macmillan Cancer Support, and leading oncologist Professor Martin Gore.
Ms Thomas said: "It's absolutely critical we understand what happened and make sure this situation never happens to another person again.
"As a joint chair of this inquiry, I am committed to ensuring the voices of people affected are heard and the lessons are learnt."
The helpline for those who think they may be affected is 0800 169 2692.
Hunt: Welsh officials do not believe breast cancer screening error affected them
The Welsh Government does not believe a breast cancer screening error has affected them even though they used the same IT system as England, according to Jeremy Hunt.
The Health Secretary outlined the Welsh administration's view in the Commons as he updated MPs about a "computer algorithm" failure dating back to 2009, which has affected the screening programme in England.
This is said to have resulted in tens of thousands of women aged 68 to 71 failing to receive an invite to their final routine screening, with Mr Hunt saying the intention is to contact all those living within the UK who are registered with a GP before the end of May.
Women aged under 72 will be automatically sent an invitation to a catch-up screening while those aged 72 and over will be given access to a helpline to receive clinical advice to decide whether or not screening is appropriate, MPs heard.
Mr Hunt confirmed work is ongoing between the UK and Welsh governments to ensure the letters are sent out to everyone.
Speaking in the Commons, Labour MP Ann Clwyd (Cynon Valley) said: "Can I ask the Secretary of State what conversations he's had with the Welsh Secretary because having long since passed the ages he mentioned, I certainly was never invited for a screening, I had to ask for one and eventually got the screening in England."
Mr Hunt replied: "We haven't had conversations at ministerial level but we have had conversations at official level.
"The Welsh administration doesn't believe that this problem has affected them, even though they were using the same IT system that we were using in England, but that is their view.
"But obviously our concern is of people living in England registered with a Welsh GP or people living in Wales registered with an English GP.
"And so that's why we're having constructive discussions to share IT information to make sure that everyone in England or Wales registered with a GP will get that letter."