Bush was speaking on Monday during a brief visit on the final leg of his Asian tour.
"Free people did not falter in the Cold War, and free people will not falter in the war on terror," the president said in an applause-filled speech to members of parliament and others at the Government House on Monday.
Bush was greeted at the Government House by flower-toting children dressed in traditional Mongolian robes, and soldiers wearing bright red, blue and yellow overcoats.
The president said Mongolia has stood with the United States as "brothers in the cause of freedom".
He called Mongolians' success in driving communist leaders from power 15 years ago an example for the world.
"Like the ideology of communism, the ideology of Islamic radicalism is destined to fail - because the will to power is no match for the universal desire to live in freedom," Bush said.
Visit a first
Bush's four-hour stop in Mongolia was the first by an American president.
The brief visit was a reward for Mongolia's pursuit of democracy and support for the US fight against terrorism. But during his tour, a bitter debate at home about the war in Iraq has followed him.
After a welcome ceremony outside Government House, Bush and Mongolian President Nambaryn Enkhbayar headed to the building's courtyard and entered a white tent known as a ger where they met privately.
Bush met Mongolian Prime
Minister Tsakhia Elebegdorj (R)
Gers are round, easily packable felt tents that are well-suited to life in Mongolia's harsh climate and nomadic culture.
Bush has been fiercely defending his Iraq policy across Asia amid growing criticism in Washington.
He appeared determined to scale back the rhetoric, and rejected the notion that it is unpatriotic to disagree with him.
"People should feel comfortable about expressing their opinions about Iraq," Bush said.
Bush brought up the growing Iraq debate when he met reporters on Sunday after inconclusive talks with China's President Hu Jintao about friction in US-China relations.
His welcome was warmer in Mongolia, which has been eager for closer military relations with the US, and has provided about 120 Mongolian soldiers in Iraq.
The number is small, but White House officials are quick to point out that, per capita, only two other countries - the United Kingdom and Denmark - have sent more of their soldiers to Iraq.
The Mongolians have been rewarded with $11 million in US aid to improve military forces.
Bush met Pacific-Rim leaders at
the recent Apec summit
Bush also noted that the country was one of 16 chosen to share in $1 billion in US aid as part of his Millennium Challenge Account that rewards poor countries that show a commitment to economic and government reform.
Bush urged the parliament to pass anti-corruption legislation as part of the transition to a successful democracy.
Mongolia's share of the $1 billion is subject to approval after the country submits a spending proposal to Washington.
The millions of dollars expected from the programme could make a big difference for a country with a total gross domestic product of only $1.1 billion.
Bush said US forces were proud to serve with the "fearless warriors" of Mongolia, home of legendary, ferocious horseman-warrior Genghis Khan.
Bush specifically thanked two Mongolian soldiers who shot and killed a bomber who was trying to drive a truck full of explosives into a multinational force mess tent in southern Iraq.
With eight more US military deaths over the weekend, Bush reminded Mongolians that their transition to liberty was not always easy. But he said Mongolians had built a better life with their struggle against communism.
"People should feel comfortable about expressing their opinions about Iraq"
US President George Bush
"Like the ideology of communism, Islamic radicalism teaches that the innocent can be murdered to serve their brutal aims," Bush said.
"Like the ideology of communism, Islamic radicalism is dismissive of free peoples, claiming that men and women who live in liberty are weak and decadent.
Mongolia was the last scheduled stop during Bush's weeklong visit to Asia, which included visits to Japan, South Korea and China.
Bush ran into stiff resistance from the Chinese to his call for expanding religious freedom and human rights.
He also reported no breakthroughs towards reducing China's massive trade surplus, overhauling its currency system or protecting intellectual property right.