In Iran - facing the possibility of U.S.-led international sanctions against Tehran for its nuclear development - the English-language Iran News said most Americans had "finally caught up with the rest of the world in rebuking the irresponsible, militaristic, arrogant, belligerent and entirely destabilizing policies of the Bush administration."
It warned against "expecting a sea change in American foreign policy," however.
"Bush's wrong strategy in the Middle East," was responsible for the Republican defeat, Iranian state television said in a commentary, which also cited corruption.
The conservative Iranian paper Siyasat-e Rouz predicted that there would be "tangible changes" toward Iran and other "opponents of America's policies," but because Bush was still in the White House, there would be no "major change" in American foreign policy, it said.
Another daily, Kayhan - which is linked to Iran's supreme leader - predicted that Bush would be obliged to take "more cautious steps."
The English-language Lebanese Daily Star said it had taken American voters six years, but now they knew like everyone else that "George W. Bush is a dangerous cowboy who needs to be restrained."
It remained to be seen, it said, "whether the rebuke delivered by American voters will be reflected in U.S. policies overseas, and there is little reason for optimism."
In Jordan, considered one of America's staunchest Arab allies in the region, newspaper editor Nabil al-Sharif said that Bush's policies are "dangerous to the region and to the world."
"We are delighted that the American voters have at least disassociated themselves from these dangerous policies," al Sharif was quoted as saying.
Other Arab and Muslim reaction was even stronger. Syrian lawmaker Suleiman Hadad said Bush "is no longer acceptable worldwide" while Pakistani opposition parliamentarian Hafiz Hussain Ahmed said the president "deserves a Saddam-like death sentence."
Officials in Afghanistan and Iraq, where Rumsfeld played a key role in the wars that changed those countries, had mixed reactions to news of his departure.
The government of Afghanistan said it was sad to see Rumsfeld resign.
President Hamid Karzai's chief of staff Jawad Ludin was quoted by wire services as saying the country was "very pleased and very grateful" for his support.
Officials said that they did not believe U.S. policy toward the country would change.
But in Iraq, lawmakers said they expected to see changes as a result of Rumsfeld's departure.
Hassan al-Sunnaid, a member of Prime Minister Nouri al-Maliki's Dawa party, said he expected to see "a major change in military commands in Iraq."
"We hope he [the new defense secretary] will be more effective and more serious in achieving security in Iraq," al-Sunnaid said.
Hassan Rhadi, a Shiite minister in al-Maliki's cabinet, said Iraq would withhold judgment on Rumsfeld's successor until it sees how he performs.
Bush has named former CIA chief Robert Gates as the new defense secretary.
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