Ten teenagers, including two 15-year-old boys, were among the victims.
Their bodies were found in a lorry trailer in an industrial park in Grays on 23 October.
Essex coroner Caroline Beasley-Murray said: "May I take this opportunity to offer my deepest condolences to the victims' families."
Assistant Chief Constable Tim Smith said: "This was an incredibly important process and our team has been working hard to bring answers to worried families who fear their loved one may be among those whose tragic journey ended on our shores.
"Our priority has been to identify the victims, to preserve the dignity of those who have died and to support the victims' friends and families."
Pham Thi Tra My, 26, sent her family a message on the night of 22 October - the day before the 39 people were found dead - saying she could not breathe and her "trip to a foreign land has failed".
Le Minh Tuan, the father of 30-year-old Le Van Ha, who comes from an agricultural part of Vietnam, previously told the BBC he was convinced his son was among the dead.
By Danny Shaw, BBC home affairs correspondent
The process of identifying those who died in the container has taken just over two weeks.
An Identification Commission, overseen by the coroner for Essex, used fingerprints, DNA, dental records and distinctive body markings such as tattoos and scars to confirm the victims' names.
Most of those who died were in their 20s and 30s; there were 10 teenagers; and two were in their early 40s. Eight were women.
All the victims came from central or northern Vietnam.
Police say the authorities are now discussing arrangements for the bodies to be repatriated.
The bodies were found on the Waterglade Industrial Estate in a container which had been shipped to nearby Purfleet from Zeebrugge in Belgium.
Lorry driver Maurice Robinson, from Northern Ireland, has appeared in court charged with offences including 39 counts of manslaughter.
Extradition proceedings have also begun against 22-year-old Eamonn Harrison, who was arrested in Dublin on a European Arrest Warrant.
Police are also seeking two brothers from Northern Ireland, Ronan and Christopher Hughes, who are wanted on suspicion of manslaughter and people trafficking.
There have been 11 arrests in two provinces of Vietnam in connection with the deaths.
The victims are:
- Pham Thi Tra My, 26-year-old woman from Ha Tinh
- Nguyen Dinh Luong, 20-year-old man from Ha Tinh
- Nguyen Huy Phong, 35-year-old man from Ha Tinh
- Vo Nhan Du, 19-year-old man from Ha Tinh
- Tran Manh Hung, 37-year-old man from Ha Tinh
- Tran Khanh Tho, 18-year-old man from Ha Tinh
- Vo Van Linh, 25-year-old man from Ha Tinh
- Nguyen Van Nhan, 33-year-old man from Ha Tinh
- Bui Phan Thang, 37-year-old man from Ha Tinh
- Nguyen Huy Hung, 15-year-old boy from Ha Tinh
- Tran Thi Tho, 21-year-old woman from Nghe An
- Bui Thi Nhung, 19-year-old woman from Nghe An
- Vo Ngoc Nam, 28-year-old man from Nghe An
- Nguyen Dinh Tu, 26-year-old man from Nghe An
- Le Van Ha, 30-year-old man from Nghe An
- Tran Thi Ngoc, 19-year-old woman from Nghe An
- Nguyen Van Hung, 33-year-old man from Nghe An
- Hoang Van Tiep, 18-year-old man from Nghe An
- Cao Tien Dung, 37-year-old man from Nghe An
- Cao Huy Thanh, 33-year-old man from Nghe An
- Tran Thi Mai Nhung, 18-year-old woman from Nghe An
- Nguyen Minh Quang, 20-year-old man from Nghe An
- Le Ngoc Thanh, 44-year-old man from Dien Chau
- Pham Thi Ngoc Oanh, 28-year-old woman from Nghe An
- Hoang Van Hoi, 24-year-old man from Nghe An
- Nguyen Tho Tuan, 25-year-old man from Nghe An
- Dang Huu Tuyen, 22-year-old man from Nghe An
- Nguyen Trong Thai, 26-year-old man from Nghe An
- Nguyen Van Hiep, 24-year-old man from Nghe An
- Nguyen Thi Van, 35-year-old woman from Nghe An
- Tran Hai Loc, 35-year-old man from Nghe An
- Duong Minh Tuan, 27-year-old man from Quang Binh
- Nguyen Ngoc Ha, 32-year-old man from Quang Binh
- Nguyen Tien Dung, 33-year-old man from Quang, Binh
- Phan Thi Thanh, 41-year-old woman from Hai Phong
- Nguyen Ba Vu Hung, 34-year-old man from Thua Tien Hue
- Dinh Dinh Thai Quyen, 18-year-old man from Hai Phong
- Tran Ngoc Hieu, 17-year-old boy from Hai Duong
- Dinh Dinh Binh, 15-year-old boy from Hai Phong