Thirteen children, aged as young as one, were among the Afghan Sikhs rescued from a shipping container on Saturday.
Members of the 35-strong group found screaming and banging after arriving at the Port of Tilbury, Essex, from Belgium, have been quizzed by officials after being released from hospital.
Kamaljit Singh Mataharu, a translator used by police, told ITV News that the group believed they would have died if they remained trapped for another 30 minutes.
He added that they told him: “It was pitch black, without any air. It soon became extremely uncomfortable.”
Mr Mataharu said there were 15 families, all from Kabul, packed into the container after they had travelled to Europe in a lorry.
Essex Police said they include nine men and eight women aged between 18 and 72, as well as 13 children aged between one and 12.
The man found dead inside after the container was opened — prompting a homicide investigation — is believed to have been in his 40s.
A postmortem was carried out yesterday but further tests are set to be undertaken to establish the cause of death.
Once police interviews have been completed, the survivors will be passed into the care of the Border Force.
Superintendent Trevor Roe, of Essex Police, said: “The welfare and health of the people is our priority at this stage.
“Now they are well enough, our officers and colleagues from the Border Force will be speaking to them via interpreters so we can piece together what happened and how they came to be in the container. We now understand that they are from Afghanistan and are of the Sikh faith.
Belgian police indicated that they believe the immigrants were probably already inside when the container was delivered to a European port before setting sail for Britain.