President Barack Obama, meanwhile, focused on "a new chapter" for Venezuela, following 14 years in which Chavez cast himself as a bulwark against U.S. domination.
Chavez, who died Tuesday at age 58, was seen as a hero by some for his socialist programs, his anti-U.S. rhetoric and gifts of cut-rate oil. Others considered him a bully who repressed his opponents.
A teary-eyed Bolivian President Evo Morales, one of Chavez's closest allies and most loyal disciples, declared that "Chavez is more alive than ever."
"Chavez will continue to be an inspiration for all peoples who fight for their liberation," Morales said Tuesday in a televised speech. "Chavez will always be present in all the regions of the world and all social sectors. Hugo Chavez will always be with us, accompanying us."
Former London Mayor Ken Livingstone, another left-wing fan of the Latin American strongman, told The Associated Press that Chavez's ability to shrug off American pressure and weather what he described as a U.S.-backed coup attempt had inspired the entire continent to defy Washington.
"The fact that he survived encouraged other Latin American nations to break free and put their own people ahead of corporate interests," Livingstone said in a telephone interview. "Before him, the governments there were just creatures of the White House. Now they are generally pursuing policies that help their own people."
Chavez pulled Venezuela out of America's sphere of influence and embraced Washington's rivals including Cuba, Iran and Russia. Officials in all three countries had effusive praise for the late leader.
Iranian President Mahmoud Ahmadinejad announced a day of mourning and compared Chavez to a saint, saying he will "return on resurrection day."
Ahmadinejad said he has "no doubt Chavez will return to Earth together with Jesus and the perfect" Imam Mahdi, the most revered figure of Shiite Muslims, and help "establish peace, justice and kindness" in the world. The Iranian leader said he believes something "suspicious" caused the cancer that killed Chavez.
In Cuba, President Raul Castro's government declared two days of national mourning and ordered flags to fly at half-staff.
"It is with deep and excruciating sorrow that our people and the revolutionary government have learned of President Hugo Rafael Chavez Frias' decease," it said in a statement read on the nightly state TV newscast. "The Cuban people view him as one of their most outstanding sons."
Some islanders worried about the loss of the country's No. 1 ally, who has sent billions of dollars of oil to Cuba at preferential terms.
"It's a very tough blow. ... Now I wonder, what is to become of us?" said Maite Sierra, a 72-year-old Havana resident.
Across Latin American, grief-stricken supporters lit candles, waved Venezuelan flags or brandished portraits of the late leader in solidarity with mourners in Venezuela.
In Moscow, candles, flowers, and a traditional Russian bottle of vodka were left outside Venezuela's embassy. Russian President Vladimir Putin called Chavez "an extraordinary and strong man who looked into the future and always aimed high," while Prime Minister Dmitry Medvedev posted photographs of himself with Chavez to his Facebook page.
Chinese President Hu Jintao, who steps down this month, and his replacement, Xi Jinping, also sent their condolences to Vice President Nicolas Maduro, the interim Venezuelan president ahead of new elections.
Venezuela took out tens of billions of dollars in Chinese loans, repayable in oil, which helped fund social programs and consumer goods giveaways that made Chavez popular. His anti-American policies and posturing also pleased some in Beijing, though Chinese leaders were careful not to show public support.
China's Foreign Ministry spokeswoman Hua Chunying called Chavez "a good friend to the Chinese people."
In the United States, Obama issued a statement reaffirming Washington's support for the "Venezuelan people and its interest in developing a constructive relationship with the Venezuelan government."
"As Venezuela begins a new chapter in its history, the United States remains committed to policies that promote democratic principles, the rule of law, and respect for human rights," the statement read.
Republican U.S. Congresswoman Ileana Ros-Lehtinen of Florida called Chavez's death "an opportunity for democracy in Venezuela."
Some of the estimated 190,000 Venezuelan immigrants living in the United States — about half of them in Florida — turned out cheering and waving their country's flag and expressed hope Tuesday that change would come to their homeland.
"We are not celebrating death," Ana San Jorge, 37, said amid a jubilant crowd in the Miami suburb of Doral. "We are celebrating the opening of a new door, of hope and change."
Several U.S. allies offered praise for Chavez, though some, like France's Socialist President Francois Hollande, noted that "not everyone shared" his political views.
German Foreign Minister Guido Westerwelle said in a statement that Chavez's death was "a heavy blow," but also said Venezuela would move on to "new times." Spanish Prime Minister Mariano Rajoy sent a telegram saying that with the death of Chavez "one of the most influential figures in Venezuela's contemporary history has disappeared."
Canadian Prime Minister Stephen Harper also sent his condolences — although he pointedly offered hopes for a "a better, brighter future based on the principles of freedom, democracy, the rule of law, and respect for human rights."
Argentine President Cristina Fernandez declared three days of mourning nationwide. She and President Jose Mujica of neighboring Uruguay prepared to travel to Venezuela for the funeral.
In Nicaragua, a nation that broadly benefited from Venezuelan cut-rate oil, Rosario Murillo, the wife and spokeswoman of President Daniel Ortega, said Chavez is "one of the dead who never die."
"We are all Chavez," she said in televised comments.
Former U.S. President Jimmy Carter released a statement saying Chavez had a vision to bring "profound changes" to those in his country who were neglected and marginalized.
"Although we have not agreed with all of the methods followed by his government, we have never doubted Hugo Chavez's commitment to improving the lives of millions of his fellow countrymen," Carter wrote.
A wistful Rafael Correa, president of Ecuador and another of Chavez's closest allies, predicted Chavez would have a lasting influence. "We have lost a revolutionary, but millions of us remain inspired."
His influence extended beyond Latin America. Nabil Shaath, an adviser to the Palestinian president, called Chavez "a loyal friend who passionately defended our right to freedom and self-determination."
There was no shortage of emotional farewells to a socialist hero who some feel rivaled the revolutionaries of the 1960s.
Cuban folk singer Silvio Rodriguez, whose ode to revolutionary Ernesto "Che" Guevara became famous, used the song's title words to bid farewell to Chavez on his blog.
"Hasta siempre, comandante," he wrote, Spanish for "Farewell forever, commander."
Orsi reported from Havana, Cuba. Associated Press staff from around the globe contributed to this report.