Most of the victims were between 20 years and 40 years old.
The Vietnamese embassy in London said it was "in deep sorrow" over the confirmation that the 31 men and eight women discovered dead in the back of a truck on October 23 were from the southeast Asian country.
They are also hunting for two brothers from Northern Ireland while Vietnamese police have arrested 10 people there in connection with the deaths.
Numerous families had previously spoken about their missing loved ones, painting a picture of young men and women desperate for a better life.
Ten teenagers, including two 15-year-old boys, were among the victims whose bodies were found on October 23 in a container in the southeastern British region of Essex. His sister had already moved to South Korea.
Hoang Van Tiep, 18, and his cousin, Nguyen Van Hung, 33, from Dien Chau district in Nghe An, left a year ago together to move to France.
Some of the names had emerged in news reports before, among them Pham Thi Tra My, a 26-year-old woman whose heartbreaking final text message to her mother vividly brought home the scale of the tragedy.
"I'm sorry Mom. My path to overseas doesn't succeed". "Mom, I love you and dad very much". I'm dying because I can't breathe.
Essex police had worked with Vietnamese authorities to identify the victims, using fingerprints, DNA, medical and dental records.
"This was an incredibly important process and our team has been working hard to bring answers to anxious families who fear their loved one may be among those whose tragic journey ended on our shores", said Assistant Chief Constable Tim Smith.
"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." a senior officer in charge of the inquiry said.
The routes across Europe can be extremely risky, particularly the final leg to the United Kingdom where many migrants are hidden inside trucks.
Several people suspected of being involved have been arrested, according to state media.
The truck driver, from Northern Ireland, has been charged with 39 counts of manslaughter, conspiracy to traffic people, and money laundering.