The Prime Minister said the perpetrators of the crime "should be hunted down and brought to justice" as police began the task of identifying the victims, including a teenager, discovered at the Waterglade Industrial Park in Grays early on Wednesday.
Police said the lorry driver, a 25-year-old man from Northern Ireland, had been arrested on suspicion of murder.
The lorry is from Bulgaria and entered the UK at Holyhead in north Wales, one of the main ports for ferries from Ireland.
Police have said tracking the route of the lorry "will be a key line of inquiry" amid concerns it may have made its way to the British mainland unchecked by avoiding the ports of Calais and Dover, which have more stringent monitoring of people smuggling.
Thurrock MP Jackie Doyle-Price said the people smugglers responsible must be caught.
She told the House of Commons: "To put 39 people into a locked metal container shows a contempt for human life that is evil.
"The best thing we can do in memory of those victims is to find the perpetrators and bring them to justice."
Haulage industry experts suggested the lorry was likely to have arrived in Ireland from Cherbourg or Roscoff, avoiding the tighter checks for people-smuggling at Calais and Dover.
Richard Burnett, chief executive of the Road Haulage Association, said: "This tragedy highlights the danger of migrant gangs people-smuggling on lorries."
He told the PA news agency: "It's highly unlikely that if this vehicle has come from Europe that it's been physically checked.
"Because of the migrant issue at Dover and Calais, you've got far more checks."
Mr Burnett said the container appeared to be a refrigerated unit, where temperatures can be as low as -25C - and described conditions for anyone inside as "absolutely horrendous".
Seamus Leheny, Northern Ireland policy manager for the Freight Transport Association, told PA: "If the lorry came from Bulgaria, getting into Britain via Holyhead is an unorthodox route.
"People have been saying that security and checks have been increased at places like Dover and Calais, so it might be seen as an easier way to get in by going from Cherbourg or Roscoff, over to Rosslare, then up the road to Dublin."
During a brief press conference at Grays police station, Deputy Chief Constable Pippa Mills said: "We believe the lorry is from Bulgaria and came into the UK through Holyhead on October 19.
"At this stage we have not identified where the victims are from or their identities and we anticipate this could be a lengthy process."
Police have set up a casualty bureau for anyone to call if they have concerns about relatives who may be involved.
Aerial footage from the scene shows forensic officers walking in and out of the lorry which is parked on Eastern Avenue opposite Pirtek and Hydraquip.
Two tents have been erected - one in front of the lorry and one behind it with police vehicles on either side.
Police have taped off the area as they investigate.
People who work on the industrial estate spoke of their shock.
Dan Peters, managing director of Pirtek, a hydraulic maintenance company, said: "We're not able to gain access to our units to open up for daily business.
"Obviously, it's terrible news."
Mr Peters said the industrial estate is made up of courier and logistics companies.
The discovery is not the worst of its kind in the UK.
The bodies of 58 Chinese people were found in a container at Dover, Kent, in 2000.
Seven men were jailed by a Dutch court for their role in the human-smuggling operation that led to the young people suffocating and the Dutch lorry driver was jailed for 14 years.