The intervention is the biggest by Western powers against Assad in the country's seven-year civil war , but the three countries said the strikes were limited to Syria's chemical weapons stock and not aimed at toppling Assad or intervening in the conflict.
Only three countries — Russia, China, and Bolivia — voted in favour of the resolution at the emergency Security Council meeting. Eight countries voted against and four abstained.
NATO said all 29 members in the alliance backed the action.
NATO Secretary General Jens Stoltenberg said the strikes were about making sure that chemical weapons were not "used without impunity," adding that the action by the allies was "very successful" and "significantly degraded the abilities of Syrian President Bashar al-Assad's forces to launch chemical attacks again."
The US ambassador to the UN said President Donald Trump told her that if the Syrian regime uses poisonous gas again "the US is locked and loaded" to strike again.
UN Secretary General Antonio Guterres urged all Security Council members to exercise restraint and avoid an escalation in Syria but condemned the use of chemical weapons.
Strikes to degrade Syria's chemical weapons capability and impede another chemical attack
In a televised address from the White House, Trump said: "A short time ago, I ordered the United States Armed Forces to launch precision strikes on targets associated with the chemical weapons capabilities of Syrian dictator Bashar al-Assad."
"These are not the actions of a man. They are crimes of a monster instead," Trump said referring to Assad and his role in the chemical weapons attacks.
As he spoke, at least six loud explosions were heard in Damascus on Saturday and smoke was seen rising over the Syrian capital.
Syrian state TV said the army's air defences shot down 13 missiles fired in a US-led attack in the Kiswah area south of the capital Damascus.
US Defence Secretary James Mattis and Marine General Joseph Dunford said three main chemical weapons facilities were targeted at 9 pm EST (1 am GMT) by missiles from both the sea and aircraft, which triggered Syrian air defences; a scientific research centre in Damascus, a chemical weapons storage facility west of Homs and a chemical weapons equipment storage and an important command post, also near Homs.
The Pentagon said there had been chemical weapons agents at one of the targets and they had "significantly crippled Syria's ability to produce such weapons." But they could not confirm how many missiles hit their targets and said no other attacks were planned.
They said the strike was designed to degrade Syria's chemical weapons capability without killing civilians or the many foreign fighters in Syria's multi-sided civil war, particularly those from Russia.
"We specifically identified these targets to mitigate the risk of Russian forces being involved," Dunford told reporters, adding the US military advised Russia of airspace that would be used in the strike but did not "pre-notify them."
"The purpose of our actions tonight is to establish a strong deterrent against the production, spread and use of chemical weapons," Trump said.
The US president had sharply critical words for both Russia and Iran, which have backed Assad's government.
"To Iran and to Russia, I ask, what kind of a nation wants to be associated with the mass murder of innocent men, women and children?" Trump said.
UK Prime Minister Theresa May said the strikes were not about "regime change," but "about a limited and targeted strike that does not further escalate tensions in the region and that does everything possible to prevent civilian casualties."
May said while the strike was targeted at Syria, it sent a message to anyone who used chemical weapons.
Britain has accused Russia of being behind a nerve agent attack on a former Russian spy in England last month, a charge Moscow has rejected.
"This is the first time as prime minister that I have had to take the decision to commit our armed forces in combat and it is not a decision I have taken lightly," she said.
"I have done so because I judge this action to be in Britain's national interest. We cannot allow the use of chemical weapons to become normalised within Syria, on the streets of the UK, or anywhere else in our world."
French President Emmanuel Macron said France had joined the US and Britain in an ongoing operation of strikes to target Syria's chemical weapons facilities.
"We cannot tolerate the recurring use of chemical weapons, which is an immediate danger for the Syrian people and our collective security.
'Red line set by France was crossed'
"The red line set by France in May 2017 has been crossed," he said.
Macron spoke to Trump and May discuss the results of the strikes.
US Defence Secretary James Mattis said the air strikes were a "one time shot" to send a strong message to Assad.
The US and its allies fired more than 100 missiles at Syria.
Mattis told reporters there were no reports of US losses in the operation.
US Joint Chief of Staff Marine General Joseph Dunford said the strike was designed to degrade Syria's chemical weapons capability without killing civilians or the many foreign fighters in Syria's multi-sided civil war, particularly those from Russia.
Strikes are 'unacceptable and lawless' said Russian Foreign Minister
Russia's ambassador to the US warned that there would be consequences for the US-led military strikes, adding that it was not acceptable to insult Russia's president.
Russia's Foreign Ministry Sergei Lavrov called the strikes "unacceptable and lawless."
Spokeswoman Maria Zakharova wrote on Facebook: "First the 'Arab spring' tested the Syrian people, then Islamic State, now smart American rockets. The capital of a sovereign government, trying for years to survive under terrorist aggression, has been attacked."
Iran's Foreign Ministry has also strongly condemned the airstrikes and warned of its regional consequences.
Syria and its allies also made clear that the attack will not do any meaningful harm to Assad.
A senior official in the regional alliance that has supported President Bashar al-Assad in the Syrian war told Reuters the sites that were targeted had been evacuated days ago thanks to a tip-off from Russia.
Inspectors from the Organisation for the Prohibition of Chemical Weapons (OPCW) were due to inspect the site of the alleged attack on Saturday. Moscow condemned the Western allies for not waiting to see their findings.
Syria has denied carrying out the chemical attack and warned that Western military strikes would risk starting a war.
In 2013, Syria had agreed to hand over all of its chemical weapons after a nerve gas attack killed hundreds of people in Douma.