The Russian vetoes have been one of the main irritants in strained relations between Washington and Moscow, with Western diplomats accusing Russia of trying to protect the Assad government.
Trump's declaration of "mission accomplished" had an awkward historic ring.
The three sites included the Barzah chemical-weapons research and development center near Damascus that was hit by 76 missiles and "destroyed".
Although officials said the singular target was Assad's chemical weapons capability, his air force, including helicopters he allegedly has used to drop chemical weapons on civilians, were spared.
Trump on Friday ordered precise strikes against Syrian military facilities with Britain and France in a coordinated action. No casualties have been reported.
"To succeed in the long run, we need a comprehensive strategy for Syria and the entire region", Republican Senator John McCain said in a statement after the latest strikes.
While he declared success, the Pentagon said the pummelling of three chemical-related facilities left enough others intact to enable the Assad government to use banned weapons against civilians if it chooses. "We have to be conscious of the fact that we can't allow even the smallest use of chemical weapons".
Despite describing the intervention as "successful", Britain's foreign minister said there were now no plans for further attacks.
He said the mission was to "establish a strong deterrent against the production, spread, and use of chemical weapons,"as he criticised Iran and Russian Federation for supporting Assad".
According to the White House, Trump and Macron affirmed in the call that the joint airstrikes were "successful and necessary" to deter Syria from any further use of chemical weapons.
And it would establish a new body to determine responsibility for chemical attacks, call for a cease-fire in Syria, unimpeded access for all humanitarian aid, and an urgent resumption of negotiations on a political settlement.
- The Russian embassy in the United States said it had warned that such actions would "not be left without consequences", adding that insulting President Vladimir Putin was "unacceptable and inadmissible". A UN Security Council vote brought by Syria's ally, Russia, and condemning the US-led strikes was rejected.
Ms Haley said the U.S. and its allies had given diplomacy "chance after chance" but Russian Federation had continued to veto United Nations resolutions.
Speaking to Rouhani by telephone the day after US-led strikes on suspected chemical weapons facilities, Putin said "if such actions, carried out in violation of the United Nations Charter, are repeated, that would inevitably provoke chaos in worldwide relations", according to a statement from the Kremlin.
Damascus: Syrian President Bashar al-Assad said Western strikes on government military installations today only made him more keen to fight back against his opponents, in comments published by his office. China too opposed the strikes.
Inspectors from the Organization for the Prohibition of Chemical Weapons (OPCW) arrived in the Syrian capital, Damascus, on April 14 and were expected to begin their investigation at the site of the alleged chemical attack on April 15.
But the only norm we're really enforcing is the one that says the United States is exempt from the laws and norms by which our adversaries must abide.
UN Secretary General Antonio Guterres told the Security Council that the airstrikes were reportedly limited to three military locations inside Syria.
The narrowly targeted pre-dawn military operation, which took aim at three alleged chemical weapons facilities, earned quick scorn from Russian Federation, but a push by Moscow for condemnation of the strikes at the Security Council fell far short.
British Tornado and Typhoon warplanes, American B-1 bombers and French Rafale jets also took part in the strikes.
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