The details were contained in a highly critical report compiled by John Vine, the chief inspector of the UK Border Agency, which will embarrass Theresa May, the Home Secretary.
Airport records for April and May last year showed that four passengers were found with small quantities of cannabis after their luggage was searched at the airport's North Terminal.
Under the rules, they should have been arrested and their case passed to an investigation team. But instead, inspectors said, all four "were allowed to proceed with a warning".
Inspectors are allowed to let low-level drugs offenders pay a fine, known as a compound settlement, which must be approved by a senior officer. In none of the four cases did the passenger pay a financial penalty, the report found.
The passengers' personal details were recorded on an internal UK Border Agency computer system but were not passed to the police or placed on the Police National Computer.
Under the 1971 Misuse of Drugs Act the maximum penalty for bringing a class B drug such as cannabis into Britain is 14 years in jail. The drug was downgraded to Class C by the former Labour Home Secretary, David Blunkett, in 2004.
Mary Brett, from Europe Against Drugs, a campaign group, told the Daily Mail: "It sends the message, 'bring your cannabis here to Britain and if we catch you it doesn't matter'.
Garry Cullen, assistant chief inspector of the UK Border Agency, admitted the disclosures were "worrying".
"Clearly we were concerned with what we found," he said.
Lucy Moreton, deputy general secretary of the Immigration Services Union, added: "Given the pressure on staff I can understand if the view was to confiscate the drugs and let the individual go."
The report, published last week, also found that detection officers who wanted to question a black passenger would also detain white people "purely to avoid potential race discrimination complaints".
It found one in seven decisions to allow a passenger entry was not supported by evidence, including some with "clear attempted deception".
A Border Force spokesman said: "Our message is clear: Importing cannabis is illegal and our officers will seize it and other illegal drugs if smugglers try to bring it into the UK.
"In recent weeks we have seized 100 kilos [22 0lb] of cannabis alone, and we are continuing to tackle the drug trade and prosecute smugglers."