Russian state media said Assad was in a "good mood" and denied speculation that he and his family had fled to Iran before Friday night's attack.
Before dawn on Saturday, a coalition of US, United Kingdom and French forces launched more than 100 missiles targeting three chemical facilities in Syria in retaliation for an apparent chemical attack that killed dozens of civilians on April 7.
The issue had been dealt with by Russian experts, said the commander, who spoke to Reuters on condition of anonymity. The legality of the air strikes under global law, however, remains questionable.
USA sources said they had obtained blood and urine samples from victims which had tested positive for chlorine and a nerve agent.
In a telephone conversation with his Iranian counterpart, Hassan Rouhani, Putin and Rouhani agreed that the Western strikes had damaged the chances of achieving a political resolution in the 7-year-old Syria conflict, according to a Kremlin statement.
"Vladimir Putin, in particular, stressed that if such actions in violation of the UN Charter continue, then it will inevitably lead to chaos in global relations", RIA news agency quoted the Kremlin as saying in a statement.
Syria and its ally Russian Federation deny any chemical attack took place - with Russian Federation calling it a "staged thing".
U.S. President Donald Trump said on Twitter, "The Syrian raid was so perfectly carried out, with such precision, that the only way the Fake News Media could demean was by my use of the term 'Mission Accomplished.' I knew they would seize on this but felt it is such a great Military term, it should be brought back".
Trump's decision to order airstrikes against Syria was a focus of the Sunday morning talk shows, with opponents questioning his tweet "Mission Accomplished!" and supporters defending the term's use.
Addressing told the council, US Ambassador Nikki Haley said the United States was confident that the military strikes had crippled Syria's chemical weapons programme.
"The American (military) knows well that going toward a wide confrontation and a big operation against the regime and the army and the allied forces in Syria could not end, and any such confrontation would inflame the entire region", Nasrallah said. "And should he use it again, the president has made it very clear that the United States is locked and loaded and ready to go".
The attacks struck at the heart of Syria's chemical weapons program, Washington said, in retaliation for a suspected poison gas attack a week ago.