The Trust said the BBC failed to ensure the students were aware of the risks involved in the trip in order to give their consent, which was judged to be a "serious failing".
And it concluded there had been "unfair treatment" of the LSE by being linked to the BBC's investigation.
For the programme, North Korea Undercover, reporter John Sweeney spent eight days in the country, joining an organised tour and pretending to be part of the student group. The trip had actually being organised by Sweeney's wife Tomiko Newson.
When the film came to light it was condemned by senior officials from the LSE who asked for the programme to be pulled, but it was screened as planned in April 2013 which led to a complaint from the father of one of the students in the group.
Although the Trust believed there was a "strong public interest" in the broadcast, it said the BBC "failed to consider a number of important issues and risks, and failed to deal with them appropriately".
In its report published today the Trust's editorial standards committee said: "The provision of information to the students who took part in the trip was insufficient and inadequate, and meant the daughter of the complainant did not possess the knowledge necessary to give informed consent."
In order to gain the appropriate paperwork, the team had used LSE address details for their visa applications. Today's report said doing so was "inappropriate and this, combined with other factors, risked linking the LSE with the trip and resulted in unfair treatment to the LSE".
It went on: "From the moment the BBC became involved in the trip to North Korea, Tomiko Newson (who was the trip organiser and tour leader) had a conflict of interest which was further compounded when she became employed by the BBC. The BBC should have ensured that there was someone independent of the programme team present to lead the trip.
Alison Hastings, who chairs the Trust's editorial standards committee, said: "Discovering stories in difficult or dangerous places is one of the BBC's greatest strengths.
"There was a real public interest in making this programme in North Korea but, in the Trust's view, the BBC failed to ensure that all the young adults Panorama travelled with were sufficiently aware of any potential risks to enable them to give informed consent.
"This was a serious failing, and the BBC is right to apologise to the complainants."
Sweeney was also accompanied by a cameraman/producer from Panorama in order to make their film and they travelled around the country on a tour given by North Korean guides, and they recorded their visit on the sort of cameras tourists would use, rather than taking specialist equipment.
The Trust found that the problems lay with the gathering rather than the broadcast of the material.
At the time of the broadcast concerns were raised about the impact the programme could have on academics working in sensitive areas around the world.
Following their return from the £2,000-a-head trip, the students were said to have received an email from a North Korean tourism official which said: "I reserve the right to make public and publish all personal data, including all your passports, to demonstrate that while we have been direct and honest with you, you have broken the DPRK (Democratic People's Republic of Korea) law."
The unnamed father who complained about the treatment of the students said the BBC failed the students and said they had been "unwitting human fodder used to fulfil John Sweeney and his wife's personal ambition to film inside North Korea".
He said the BBC had "risked the lives of these students in an ill-conceived scheme of questionable journalistic value", and expressed concern about the effect the filming may have had on the unwitting North Korean guides and their families.
The father said: "Absurdly, the footage from North Korea used in the broadcast added next to nothing to the public's understanding of the situation inside North Korea, given the tourist access was so tightly stage-managed by the regime.
"On any objective view given the risks involved, the deception of these students, and the use of licence fee-payers' funds to make the programme, including paying for two further 'phantom student' places, must be highly questionable.
"As a parent it is worrying to now also learn that the BBC had no exit plan for the students, and that senior BBC executives approved a strategy for the BBC team, if detained, to abandon the students and, in the words of the Trust, leave them 'in effect, as a group of young adults from a variety of different countries, all personally responsible for trying to extricate themselves from possible detention'.
"Rather than acknowledging the various BBC failures and apologising, senior executives attempted to defend the Corporation's involvement by smearing my daughter and her conduct and making a number of misleading and inaccurate statements.
"I am amazed that the Trust similarly 'concluded that there was some evidence that BBC staff had made statements about the trip ... that were subsequently shown not to have been entirely correct' but held that the senior BBC executives were simply 'interviewees', which meant the BBC was only responsible for its executives' misstatements if the editors of the programmes on which they were interviewed, knew the content to be inaccurate. This failure to hold senior executives to account is worrying.
"However, I am pleased that the BBC has today, at last, issued an apology to my daughter and me. I look forward to a broadcast correction of both this apology and all the misleading and inaccurate statements made by BBC executives - including those made by the director-general during their attempts to defend this ill-advised trip."
BBC News said it accepted the Trust's decision and said it had apologised to the LSE and the student - referred to as Student X - whose father had complained. But it said that at the time the programme was being made, it believed they were being treated "fairly".
In a statement, BBC News said: "We are pleased that the Trust found that there was a clear and strong public interest in commissioning and broadcasting the programme and that the correct referral procedures and processes were followed by the programme team and senior management.
"We also accept, however, that aspects of the BBC's handling of the project fell short in a number of areas, with the Trust finding against the BBC on four of its 21 rulings.
"In particular we have apologised to Student X for the finding by the Trust that insufficient information was given to her ahead of the trip about the involvement of the BBC journalists and the potential risks, which meant that Student X did not have sufficient knowledge on which to give informed consent. We have also apologised to the LSE for the Trust's finding that the programme created the risk of harm to the LSE's reputation.
"The Trust recognised that this programme involved a number of finely-balanced editorial judgments and that the BBC spent considerable time evaluating the risks in circumstances which were highly unusual. In the planning for and making of the programme, BBC News believed that it was treating all the students and the LSE fairly."
During its investigation, the committee was told that the BBC's risk assessment had determined that if the team was discovered, they would try to "separate" themselves from the students but this could have led to them being more vulnerable.
"The committee considered that, while this might well have been a tactic aimed at isolating the students from further risk, it would inevitably have had the effect of preventing Tomiko Newson acting primarily as group leader, and that the students would have been left, in effect, as a group of young adults from a variety of different countries, all personally responsible for trying to extricate themselves from possible detention," the report said.
It concluded that it was not viable for her to lead the trip and be certain of being able to act in the best interests of the students in North Korea from the moment the BBC became involved.
But her relationship with Sweeney was not itself a conflict of interest.
Director of the LSE, Professor Craig Calhoun, said: "LSE welcomes the finding of the Editorial Standards Committee and the letter of apology issued to the school by the BBC Executive.
"LSE would like to confirm its strong support for the production of programmes in the public interest and for journalists working to highlight important issues in dangerous parts of the world."