From Paris to Pakistan, politicians, analysts and ordinary citizens said on Wednesday they hoped the Democratic takeover of at least one house of Congress would force Bush to adopt a more conciliatory approach to the globe's crises, and teach a president many see as a "cowboy" a lesson in humility.
But there were also fears that a power split and a lame-duck president might stall global trade talks or other initiatives where US co-operation is needed.
The prospect of a sudden change in American foreign policy could also be troubling to US allies such as Britain, Japan and Australia, which have thrown their support behind the US-led invasion of Iraq.
Ali Al-Dabbagh, Iraqi government spokesman, said: "This is an internal issue for the United States. We are dealing with an administration, not persons. We are committed to an understanding with the administration."
On the comment by US President George Bush that Iraq was not "going well enough, fast enough", he said: "We feel the same, that things are not going fast enough on the security level. There should be more co-ordination, there should be more say for Iraqis.
"We think it's possible to have an improvement in Iraq."
Seeking to reassure Iraqi leaders there would be no major policy change after the Republicans lost control of the House of Representatives, envoy Zalmay Khalilzad said: "The president is the architect of US foreign policy."
Iraqi leaders say they need more US help to quell mounting sectarian violence, although polls suggest most Iraqis would like to see foreign troops leave soon.
Khalilzad told a reception attended by government officials, Iraqi legislators and embassy personnel: "He [Bush] is the commander in chief of our armed forces. He understands what's at stake in Iraq.
"He is committed to working with both houses of the American Congress to get support needed for the mission in Iraq to succeed."
In an extraordinary joint statement, more than 200 Socialist members of the European Parliament hailed the American election results as "the beginning of the end of a six-year nightmare for the world" and gloated that they left the Bush administration "seriously weakened".
In Paris, expatriates and French citizens alike packed the city's main American haunts to watch results overnight and early on Wednesday, with some standing to cheer or boo as vote tabulations came in.
One Frenchman, teacher Jean-Pierre Charpemtrat, 53, said it was about time US voters figured out what much of the rest of the world already knew.
"Americans are realising that you can't found the politics of a country on patriotic passion and reflexes," he said. "You can't fool everybody all the time - and I think that's what Bush and his administration are learning today."
For his part, Venezuela's leftist leader Hugo Chavez on Wednesday urged Bush to follow the example of his defence secretary and step down.
"The president should step down on moral grounds," he said in Caracas shortly after Bush announced the resignation news.
Chavez said: "It would be a good solution not only for the United States but also because of the huge tensions there are in the world.
"I think Bush has fallen ... the collapse is starting, heads are starting to roll, fortunately peacefully; now is the time to step down."
He said the US mid-term election was "a vote against the war, against Iraq. Hopefully the Democratic party will make the most of it," Chavez said, expressing hope Democrats would better understand the situation in Latin America.
Passions were even higher in Pakistan, where Bush is deeply unpopular despite billions in aid and staunch support for President General Pervez Musharraf.
One opposition lawmaker, Hafiz Hussain Ahmed, said he welcomed the election result but was hoping for more. Bush "deserves to be removed, put on trial and given a Saddam-like death sentence", he said.
But while the result clearly produced more jubilation than jitters around the world, there were also some deep concerns.
In Denmark, Anders Fogh Rasmussen, the prime minister, told broadcaster TV2 he hoped the president and the new Congress would find "common ground on questions about Iraq and Afghanistan".
"The world needs a vigorous USA," he said.
There was also some concern that Democrats, who have a reputation for being more protective of US jobs going overseas, will make it harder to achieve a global free trade accord.
And in China, some feared the resurgence of the Democrats would increase tension over human rights and trade and labour issues. China's surging economy has a massive trade surplus with the US.
"The Democratic Party ... will protect the interests of small and medium American enterprises and labour and that could produce an impact on China-US trade relations," Zhang Guoqing of the state-run Chinese Academy of Social Sciences said in a report on Sina.com, one of China's most popular internet portals.