"The historic resolution we just adopted enshrines our shared commitment to the goal of a world without nuclear weapons," Mr Obama told the Security Council after the resolution was adopted.
He said the next year would be "absolutely critical in determining whether this resolution and our overall efforts to stop the spread and use of nuclear weapons are successful".
UN Secretary General Ban Ki-moon called the resolution "a fresh start toward a new future".
The resolution does not specifically mention countries by name, such as North Korea and Iran, but reaffirms previous Security Council resolutions relating to their nuclear plans.
Jonathan Marcus, BBC News, New York
The unanimous backing for the US-drafted resolution is a measure of the growing sense of urgency.
There is a growing fear amongst disarmament experts both inside and outside government that the whole machinery intended to prevent the spread of nuclear weapons - whose cornerstone is the Nuclear Non-Proliferation Treaty or NPT - is looking increasingly fragile.
The NPT agreement is up for revision in May and the new consensus at the UN Security Council sends a powerful signal that this key agreement must be bolstered.
Iran's nuclear programme has been criticised by the US and five nations who are set to hold talks next week.
Iran says its nuclear ambitions are for peaceful energy purposes, but others fear it is developing weapons.
Israeli Prime Minister Benjamin Netanyahu reflected those fears in his speech to the UN General Assembly on Thursday, saying that stopping Iran acquiring nuclear weapons was the world's most urgent task.
After the resolution was passed, Iran rejected allegations about its nuclear programme as "totally untrue" and reiterated its "readiness to engage in serious and constructive negotiations with interested parties".
The resolution commits member nations to work toward a world without nuclear weapons, and endorses a broad framework of actions to reduce global nuclear risks.
It also urges states to:
join and comply with the Nuclear Non-Proliferation Treaty (NPT)
refrain from testing nuclear weapons and ratify the Comprehensive Test Ban Treaty (CTBT)
ensure safeguards of nuclear material and prevent trafficking
Mr Obama stressed that the US would play its part, seeking a new strategic arms reduction treaty with Russia and moving ahead with ratification of the test ban treaty.
"Although we averted a nuclear nightmare during the Cold War, we now face proliferation of a scope and complexity that demands new strategies and new approaches," Mr Obama said.
"Just one nuclear weapon exploded in a city, be it New York or Moscow, Tokyo or Beijing, London or Paris, could kill hundreds of thousands of people."
Also on Thursday, a UN ministerial conference adopted a declaration urging compliance with the CTBT, which has been signed or ratified by 100 countries since 1996.
US Secretary of State Hillary Clinton presence at the meeting marked the first US participation at the biannual conference since 1999, when the US Senate refused to ratify the treaty.
On Wednesday, the first day of the UN General Assembly, Iran's plans came under fire from several world leaders, including French President Nicolas Sarkozy and UK Prime Minister Gordon Brown.
Russia signalled that it might be prepared to soften its opposition to sanctions against Iran over its nuclear plans, though China, another Security Council member, said increasing pressure on Iran would not be effective.
But in a statement circulated outside the Security Council on Thursday, Iran said France and the UK had not complied with their own nuclear disarmament obligations, and were therefore not in a position to judge others.
It accused Mr Sarkozy of making "preposterous" claims, and said the UK "deliberately and cynically [had] ignored its legal commitments" to the NPT.
"Our commitment to non-proliferation remains intact," the statement said.
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