Tilbury is owned by Edinburgh-based Forth Ports.
The man was pronounced dead at the scene while 34 others, including seven children, were taken to hospital to be treated for what was described as severe dehydration and hypothermia. All are believed to be from the Indian sub-continent.
Dock workers made the discovery when they heard screaming and banging from the 40ft long container while unloading the ship yesterday morning.
The stowaways, presumed to have entered Britain illegally, are believed to have spent up to 12 hours in the container. Such shipping containers are almost airtight when sealed. Forth Ports also owns Edinburgh's Leith port, plus Grangemouth, Dundee, Rosyth, Methil, Burntisland and Kirkcaldy.
Superintendent Trevor Rowe confirmed adults and children were found in a container that arrived on a P&O ferry from Zeebrugge in Belgium at about 6.30am.
Rowe said: "All we know at the moment is that we believe them to come from the Indian sub-continent but it is still early days.
"It is a homicide investigation from the police point of view at this time."
Rowe added: "This is a humanitarian issue and the welfare of these patients is a priority."
He said checks on the other 50 containers on the P&O vessel were also carried out.
Belgian police, who are working with Essex Police, believe they have identified the lorry that delivered the container in Zeebrugge.
A trawl of CCTV footage in Belgium was also under way in an effort to find out more about where the group came from.
The UK Borders Agency had also been contacted.
The P&O freight ship, Norstream, left Zeebrugge at 10pm on Friday and arrived at Tilbury at 6am yesterday. It is not yet known where the container, one of 64 on the vessel, originated.
South Basildon and East Thurrock MP Stephen Metcalfe described the incident as tragic.
The Conservative added: "The fact that so many people appear to have travelled so far and are so desperate to get into the UK, either on their own or being trafficked, is really very sad."
He said it was vital "to get to the root causes of what is motivating people to go to such extreme lengths to travel from other parts of the world to get into the UK".
A police officer at the scene said they were still working to determine the nationalities of those on board the container but they are believed to possibly be from the Punjab.
A decontamination zone was immediately set up around the area where the people were found. However, a spokeswoman for Public Health England said there was no sign of any risk from infectious diseases such as ebola.
She said: "If it was ebola, healthcare professionals are so alert at the moment to signs and symptoms that, should there have been anyone who was showing symptoms, we would have been notified immediately. I think we can be confident that we are not dealing with that."
Basildon University Hospital sealed off its A&E department as a decontamination zone and officials said staff were responding to a major incident.
A spokesman for the East of England Ambulance service said: "We sent seven ambulances, two rapid response cars, a patient transport services vehicle, two duty operational managers, two doctors and our hazardous area response team. After initial treatment by ambulance crews, all patients have now been conveyed to surrounding hospitals for further care."
A spokeswoman for Forth Ports said the incident was a police matter, and the company had no comment to make at this stage.