The 31-year-old was the fourth migrant to die this year in the area around the port, a launchpad for attempts by migrants to smuggle across the Channel to England.
He was buried under several tonnes of paper rolls bound for a British customer after the driver lost control of the truck and it overturned, regional security officials said.
Last week, an Afghan migrant died after being run over on the road leading to the port.
The number of fatalities has however fallen dramatically since 2016 when 14 migrants died while trying to reach Britain - a popular destination for Afghans and east Africans particularly.
Migrant numbers in Calais have dwindled to a trickle after the authorities late last year bulldozed a settlement on the outskirts of the port where thousands of people had been sheltering and bussed them to shelters around the country.
Since then the police has relentlessly pursued young Africans and Asians who try to set up camp in the area, drawing sharp criticism from rights groups.
Police believe the man in his twenties fell from a shack he had been staying in on some sort of ledge above the road near the town of Roquebrune-Cap-Martin.
At least 15 migrants have died in southern France since September 2015, according to a count kept by AFP news agency.
The latest victim there likely came via the Italian border town of Ventimiglia, which has become a launching pad for migrants who make it to Italy and seek to head north into France.
Increased border patrols have made it much harder for them to do so, so many are now braving the snow-clad Alps further north in efforts to evade immigration police and reach France from Italy.
They often take treacherous high-altitude passes that locals warn will lead to deaths.
President Emmanuel Macron has adopted a two-pronged approach to migrants, cracking down on those labelled economic migrants because they are fleeing poverty rather than danger while promising to open up new avenues for refugees to travel to France legally.
In July he promised to find temporary shelter for all those on the streets by the end of 2017 - a goal he looks nearly certain to miss, with many still sleeping rough around the country.