The blaze broke out at 5am and the victim was trapped by the flames, while 15 people were evacuated from the four-storey building, Vaud cantonal police said in a news release.
Police identified the victim as an Italian man in his early 20s from Genoa.
Citing local sources, Italian news agency ANSA said he was the 21-year-old son of Gianni Onorato, the CEO of MSC Cruises, the world's largest privately-owned global cruise line, based in Geneva.
A 20-year-old man jumped from a window on the third floor of the building, landing on mattresses piled up by neighbours to break his fall, police said.
The building in the Rochesgrises neighbourhood suffered significant damage from the blaze that was brought under control by fire fighters by 8.30am.
The structure consisted of 20 apartments, some of which were occupied year round while others were used as holiday homes, police said.
The building is near the ski resort of Villars-Gryon in the Vaud Alps.