Essex Police said they had worked with Vietnamese police and the coroner to identify the 31 men and eight women.
The force said the victims’ families had been informed.
“This is an important step in the investigation and enables us to work with our Vietnamese Police colleagues to support the families of those victims,” said Assistant Chief Constable Tim Smith.
“It is only right that we provide an opportunity for family members to take in the news confirming the death of their loved ones before releasing any further information. Our thoughts remain with the families and friends of those whose tragic journey ended on our shores.”
British police have charged the driver of the lorry, 25-year-old Mo Robinson from Northern Ireland, with 39 counts of manslaughter, conspiracy to traffic people, conspiracy to assist unlawful immigration and money laundering.
Eamonn Harrison, 22, was arrested in Ireland on a British warrant and appeared at Dublin High Court on Friday charged with 39 counts of manslaughter, along with human trafficking and immigration offences, and was remanded in custody.
Essex Police have started extradition proceedings to bring him to the UK.
Detectives have also urged Ronan Hughes, 40, and his brother Christopher, 34, said to have links with the road haulage and shipping industries, to hand themselves in.
The pair, from Armagh in Northern Ireland, are wanted on suspicion of manslaughter and human trafficking.
Three other people have been released on bail pending further investigation.
Vietnamese police have arrested two people in Ha Tinh province and nine in Nghe An province on suspicion of organising people smuggling.
The victims came from six provinces – Haiphong, Hai Duong, Nghe An, Ha Tinh, Quang Binh and Hue – Vietnam’s Ministry of Public Security said in a statement.