Mr Chavez had been seriously ill with cancer for more than a year, undergoing several operations in Cuba.
Crowds of supporters gathered outside the Caracas hospital where he died, chanting “We are all Chavez!”
Following his death, Vice-President Nicolas Maduro will assume the presidency until an election is held within 30 days, Foreign Minister Elias Jaua said.
“It is the mandate that Comandante President Hugo Chavez gave us,” Mr Jaua told state television, adding that Mr Maduro would also be the candidate of the governing United Socialist Party (PSUV).
The presidential election will take place within one month.
Iran President Mahmoud Ahmadinejad sent a condolences message to Maduro.
“the spirit of a great man from a justice seeking world flying the sky, and plunged the world in grief. Venezuela lost its bravest and courageous child, and the world lost a wise and revolutionary leader,” Ahmadinejad said in his message.
Ahmadinejad condolences this tragic event to all nations and especially to “the great people of Venezuela and Chavez family.”
“he was in fact the martyr of righteous way protecting human values and the revolution to serve people of Venezuela.”
Seven days of national mourning have been declared in Venezuela and his body will lie in state until a funeral on Friday.
Iranian government also declared one day national mourning and president Ahmadinejad is to take part in his funeral.