Part of the reason is continuity. Challenger Mitt Romney is a little-known figure internationally with scant foreign policy experience, while Obama was seen even by most critics as a steady hand following a predictable course.
If he hasn't brought peace to the world's fire zones, or done much to slow climate change, or sparked global economic growth, he is credited at least with having started no new wars, and having tried to heal relations with the Muslim world even while aggressively pursuing al-Qaida and its affiliates.
"I think it is good that Obama won," said Pawel Kukiela, a 30-year-old company consultant in Poland, one of the few countries around the world where Romney has sizeable support. "He has a good program and I think it will be much better for Obama to continue what he has started."
A BBC survey during the run-up to the election found remarkable support for an Obama second term. More than 21,000 people in 21 countries were questioned in July, August and September, with residents in all but one country backing Obama. Only Pakistan, where Obama's heavy reliance on drone strikes has been unpopular, preferred Romney.
An Iraqi army officer in the capital, Baghdad, praised Obama for withdrawing U.S. troops from Iraq and planning to do the same in Afghanistan.
"These show that Obama differs from other American presidents in his foreign policy," said the officer, who gave his name as Abu Hussein.
Praise for Obama was not universal. Pakistani Taliban spokesman Ahsanullah Ahsan said Muslims should not express happiness about Obama's victory.
"We must remember that they are the enemies of Islam, and it is our duty to fight them," Ahsan told The Associated Press by telephone from an undisclosed location.
Venezuelan President Hugo Chavez, a frequent critic of U.S. policy, did not comment immediately, but the state-run Venezuelan News Agency said Obama returns to power "with various promises unfulfilled," including what it described as his failures to close the prison at Guantanamo Bay, Cuba, and set up a system to limit greenhouse gas emissions.
But the list of countries with a strong Obama preference in the BBC survey were as diverse as Nigeria, Panama, South Korea, Germany and Brazil.
A separate French poll showed broad support for Obama even from those who identified themselves as supporters of the French far-right candidate Marine Le Pen.
Jocelyne de Letrain, 60, was among Parisians who cheered Obama's win.
"I don't think that Europe would have had a good relationship with Mitt Romney," she said. "It would have been two different points of views, two different planets."
In China, Obama's re-election was good news for people concerned about Romney's vow to label China a currency manipulator. Some feared that would ignite a trade war between the world's two biggest economies.
"His re-election is in line with what the Chinese people want," said Hong Zihan, a graduate student watching the results at a U.S. Embassy event in Beijing.
In Myanmar, which is opening to the world after five decades of isolating military rule, some said they were relieved Obama was re-elected because he had chosen to engage rather than sanction their country.
"President Obama is very flexible," said Thit Oo, a 42-year-old car mechanic, "and international relations have improved during his term."