Assad and his close ally, Russia, have denied government forces ever used such weapons.
The bombing represents a major escalation in the West's confrontation with Mr. Assad's superpower ally Russian Federation, but is unlikely to alter the course of a multi-sided war which has killed at least half-a-million people in the past seven years.
The Trump administration also pushed back on comments made Sunday by French President Emmanuel Macron, who in an interview on French TV noted that President Donald Trump days earlier expressed his eagerness to withdraw USA troops from Syria, and "we convinced him to remain".
Putin denounced the strikes on Saturday "with the utmost firmness", describing them as "an act of aggression against a sovereign state which is at the forefront of the fight against terrorism".
His comments appeared in line with those of USA ambassador to the United Nations, Nikki Haley, who said at an emergency meeting of the U.N. Security Council that Trump told her that if Syria uses poisonous gas again, "The United States is locked and loaded".
The comments came as the global community was still grappling with the aftermath of the Friday's airstrikes against the Syrian government in retaliation for a suspected chemical-weapons attack that killed dozens of people.
A Kremlin statement said Mr Putin and Mr Rouhani agreed that the Western strikes had damaged the chances of achieving a political resolution in the multi-sided, seven-year conflict that has killed at least half a million people. "The humanitarian intervention is a legally debatable concept at the present time", he said in an interview with the BBC on Sunday.
The Russian defence ministry was quoted as saying that Syria deployed Russian-made surface-to-air missiles, including S-125, S-200, 2K12 Kub and Buk to repel the attacks.
They quoted the legislators as saying Assad was in a "good mood" and accepted an invitation to visit the Siberian region of Khanty-Mansi.
Trump faced criticism for using the term Saturday on social media, with critics noting that it echoed former President George W. Bush's declaration shortly after the USA invasion of Iraq in May 2003, and that the Iraq War continued for eight years. No casualties have been reported. A sarin attack triggered Trump's decision a year ago to strike a Syrian air base.
Prime Minister Shinzo Abe said the Japanese government "supports the resolve of the United States, Britain and France".
Russian Federation also decried the US -led operation as a failure, saying the majority of the rockets fired at Syria were intercepted by the Syrian government's air defense systems.
The Syrian Foreign Ministry has condemned the US -led attack, saying it violated the global law and the UN Charter, Syrian state news agency SANA reported on Saturday. The operation targeted three Syrian chemical weapons facilities - one in the capital of Damascus and two others near Homs, near the border with northern Lebanon.
"Egypt totally rejects the use of any internationally prohibited weapons on Syrian territories, demanding a transparent global investigation", the statement read.
Negotiations on the proposals are due to start on Monday, a day after a delegation from the Organisation for the Prohibition of Chemical Weapons were due to arrive in Douma. Moscow condemned the Western states for refusing to wait for their findings.
The biggest town in Eastern Ghouta, Douma, is where suspected poison gas was dropped on civilians during fighting on April 7, killing about 40 men, women and children.