The head of the UK border force Brodie Clark and two other officials have been suspended after border guards were told not to bother checking fingerprints and other personal details against a Home Office database of terror suspects and illegal immigrants.
Mrs May told MPs those responsible would be punished as she announced a series of inquiries into the scandal.
Mrs May told MPs that while ministers had started a pilot project "targeting intelligence-led checks on higher-risk" passengers, Mr Clark "authorised the wider relaxation of border controls without ministerial sanction" this summer.
"As a result of these unauthorised actions, we will never know how many people entered the country who should have been prevented from doing so after being flagged by the warnings index," she said.
Under the pilot scheme, European children, "travelling with their parents or as part of a school group, would be checked against the warnings index - designed to detect terrorists and serious criminals - when assessed by a border force official to be a credible risk", Mrs May said.
"The pilot also allowed, under limited circumstances, border force officials the discretion to judge when to open the biometric chip - which contains a second photograph and no further information - on the passports of EEA nationals.
"Those circumstances were that the measures would always be subject to a risk-based assessment, that they should not be routine, and that the volume of passengers would be such that border security would be stronger with more risk-based checks and fewer mandatory checks than with more mandatory checks on low-risk passengers and fewer risk-based checks for high-risk passengers.
"The advice of security officials was sought and they confirmed that they were content with the measures."
But on Wednesday last week, Mr Clark confirmed to Rob Whiteman, the UKBA chief executive, that "border controls had been relaxed without ministerial approval".
Biometric checks on European nationals and warnings index checks on children from the EU "were abandoned on a regular basis, without ministerial approval", Mrs May said.
Adults were not checked against the warnings index at Calais, without ministerial approval, and the fingerprints of non-EEA nationals from countries that require a visa was stopped, again without ministerial approval, she said.
"I did not give my consent or authorisation for any of these decisions," Mrs May told MPs.
"Indeed I told officials explicitly that the pilot was to go no further than we had agreed."
The pilot scheme, which had been due to end on Friday, was suspended when the unauthorised relaxation of the control was confirmed on Thursday, Mrs May said.
Graeme Kyle, the border force's director of operations at Heathrow, and Carole Upshall, director of the border force's south and European operations, were also suspended from duty "on a precautionary basis", she said.
Dave Wood, head of the UKBA enforcement and crime group, will carry out an investigation into "exactly how, when and where the suspension of checks might have taken place" and Mike Anderson, director general of immigration, is looking at the actions of the wider team working for Mr Clark.
John Vine, the independent chief inspector of the UKBA, will also conduct "a thorough review to find out exactly what happened across UKBA in terms of the checks, how the chain of command in border force operates and whether the system needs to be changed in future", Mrs May said.
"For the sake of clarity, I am very happy for Mr Vine to look at what decisions were made and when by ministers."
Reports are expected by January.
Mrs May went on: "The pilots run by the UK border force this summer were designed to improve border security, by focusing resources at passengers and journeys that intelligence led officers to believe posed the greatest risk.
"The vast majority of those officers are hard-working, dedicated public servants.
"Just like all of us, they want to see tough immigration controls and strong enforcement.
"But they have been let down by senior officials at the head of the organisation who put at risk the security of our border.
"Our task now is to make sure that those responsible are punished and to make sure that Border Force officials can never take such risks with border security again.
"That is what I am determined to do."
Mrs May was heckled in the Commons when she highlighted how border controls had been relaxed under the previous Labour government.
Shadow home secretary Yvette Cooper told MPs that UKBA staff told her today that "the shredders are on and there is a ban on internal emails".
Ms Cooper said: "The truth is that instead of strengthening the checks year on year, as all previous ministers had committed to do, this Home Secretary decided to water them down as official Government policy, even though she never told this House.
"She has blamed officials for relaxing the checks further than she intended. But she gave the green light for weaker controls.
"She said she wanted reduced checks. She claims in her statement that not checking the biometric chip should not be routine.
"Yet the interim operational instruction she refused to publish says 'We will cease routinely opening the chip within EEA passports, checking all EEA nationals under 18 against the warnings index... If for whatever reason it is considered necessary to take further measures, local managers must escalate to the Borderforce duty director to seek authority for their proposed action'.
"And look how far they went."
David Cameron has full confidence in Mrs May, who has not offered to resign as a result of the controversy, the Prime Minister's official spokesman said earlier.
The spokesman revealed that Mr Cameron was not informed of the decision to relax the checks over the summer, saying it was an operational decision to be taken within the department.
"He is obviously very concerned about these issues," the spokesman said.
"There is clearly now an investigation under way, looking into precisely what happened."
Asked whether the Prime Minister was aware of the decision to relax checks, the spokesman said: "On that kind of operational decision it is normal for departmental ministers to have oversight."
He added that neither Mrs May nor Immigration Minister Damian Green had offered to resign over the issue.
Deputy Prime Minister Nick Clegg said the situation was of "great concern".
"I have got lots of questions, a lot of people do," he told BBC Radio 4's The World At One.
"It is going to be looked at and investigated properly in an independent way."
The Liberal Democrat leader insisted the allegations had "nothing to do with" the coalition's public spending cuts, despite claims of staff shortages.
The relaxation of border controls is thought to have allowed hundreds of thousands of people to enter Britain without proper checks.
Mrs May is facing Labour demands to disclose whether any terror suspects are believed to have entered the country after border guards were instructed not to carry out certain passport checks.
The Public and Commercial Services union (PCS) has claimed that border controls were relaxed to keep queues down despite cuts to personnel. It also said the decision was authorised by ministers.
Sue Smith, of the PCS, blamed what she claimed had been a 10% reduction in border force staff.
"The travelling public understandably want to have a fast and efficient service, and yet we are also under a reduced workforce," she said.
"So, I think senior managers have seen this as a way to provide the public with what they want."
She added that senior managers had told the union that the changes to border checks had been made with the authorisation of ministers.
"As far as our staff were concerned, this was all done with ministerial authority, and that's the information we have received," she said.
But Lucy Moreton, of the Immigration Service Union, acknowledged that senior management at the UK Border Agency may have taken the decision themselves.
Speaking in the Commons, Mr Green told MPs that 5,200 staff would be cut from the UK Border Agency, taking its numbers down to 18,000, by 2015.
But he said the reductions were "not affecting the front line", saying it was the "way we use people that makes our borders more secure".
"It's important to have intelligent border controls using technology, putting the right people in the right places, so we can keep our border secure," he said.
Labour's shadow immigration minister Chris Bryant told the Commons that figures showed a total of 1,552 jobs would go in the UK border force before the next general election, including 886 this financial year.
An inquiry has been set up under John Vine, chief inspector of the UK Border Agency.
PA
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