Vietnam's Ministry of Public Security also confirmed Thursday that all 39 were from the country, noting they hailed from six provinces.
British authorities have been working with Vietnamese police and the coroner to identify the bodies found October 23in an industrial park in the town of Grays.
British police have charged two men with manslaughter as well as human trafficking and immigration offenses.
In a letter to the victims' families obtained by AFP, Vietnam's Prime Minister Nguyen Xuan Phuc said the tragedy had caused them and Vietnamese people around the world "endless pain".
Formal identification has now been completed, and the victims' families have been notified.
Assistant Chief Constable Smith said: "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".More news: Person sought in case of UFC fighter’s missing stepdaughter
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The British police initially said the bodies were believed to be Chinese.
So far, two people have been charged in the UK. "Our thoughts remain with the families and friends of those whose tragic journey ended on our shores".
Vietnam police have arrested 11 people suspected to be involved in the smuggling of the victims.
The Essex police also worked with National Crime Agency and European law enforcement officials to identify the bodies in the lorry.
Another Northern Irishman, Eamon Harrison, has been charged with manslaughter and human trafficking offences while police are still hunting for brothers Ronan and Christopher Hughes, both from Co.
Meanwhile, 23-year-old Eamonn Harrison, from Mayobridge in Co Down, faces extradition from Ireland to the United Kingdom to face charges over the discovery of the bodies.
The Vietnamese government strongly condemns acts of trafficking and calls on countries in the region and the world to step up cooperation, and resolutely prevent and combat this particularly unsafe crime so as not to repeat the same tragedies, and complete investigations soon and bring the criminals to justice.