Syrian state media has claimed that a USA missile attack on a government air base has killed seven people, including four children. It said several others were injured.
President Donald Trump authorized Thursday's strikes in reaction to the Syrian government's use of chemical weapons on civilians earlier this week.
In the base itself, six military personnel were killed.
"There are understandings in place" despite rhetoric intended for the media, he said.
Council members will hold an open meeting today and hear a briefing on the U.S. military action, the USA mission to the United Nations said.
The Syrian army said the strikes made the USA a "partner" of the Islamic State, Jabhat al-Nusra and other terrorist organizations.
Turkey, a key regional player and partner in the fight against ISIS, said it considered the USA action "a positive response to the Assad regime's war crimes", Turkish presidential spokesman Ibrahim Kalin said in a statement.
Tillerson is slated to arrive in Moscow on Tuesday for a series of talks with officials, including Russian Foreign Minister Sergey Lavrov.
The U.S. also notified its partner countries in the region prior to launching the strikes.
"It's a serious matter, it requires a serious response", he said.
Turkish President Recep Tayyip Erdogan, a longtime foe of Assad, also called for further action.
Erdogan's spokesman Ibrahim Kalin earlier said the strike against the Sharyat airbase in Homs, northern Syria, was "a positive response" to the "war crimes" of the regime of Assad.
"God bless Trump", said Abu Ali, in his forties, hours after the United States launched a barrage of cruise missiles at a Syrian government air base in the central Homs province.
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Satellite imagery suggests the Shayrat air base that was struck in western Syria is home to Russian Special Forces and military helicopters, part of the Kremlin's effort to help the Syrian government fight Islamic State and other militant groups. The missiles targeted aircraft and aircraft shelters, ammunition, air defense systems and radars.
"As always, the USA took extraordinary measures to avoid civilian casualties and to comply with the Law of Armed Conflict. Last night's strike was aimed at a different objective", he said in a statement.
Russian President Vladimir Putin has condemned the missile strikes as an illegal move that would hurt U.S. -Russia ties.
"But it shows a new attitude from the USA administration in the region which is to take initiatives individually if needed". "Washington's move impairs the Russian-U.S. relations, which are in a deplorable state, substantially", Peskov added, according to TASS.
"I think this targeted attack aims at preventing the Assad regime from using chemical weapons again and ensure that it never again comes to such awful crimes against humanity", Foreign Ministry spokesman Sebastian Fischer said.
Allowing Tuesday's attack to go unpunished would have condemned Syria to many more years of war, and while these strikes alone will not end the conflict, they offer hope that the global community, with the leadership of the United States, might just act to stop the slaughter.
The United States fired a barrage of cruise missiles into Syria in retaliation for this week's gruesome suspected chemical weapons attack against civilians, USA officials said.
The attack was the first conventional assault on another country ordered by the new president. Secretary of State Rex Tillerson, in Florida with Trump, said Moscow had failed in living up to a 2013 agreement that was meant to strip Syria of its chemical weapons stockpiles. France, Italy, and Israel also welcomed the strikes.
Meanwhile, Florida Sen. Marco Rubio, who has been one of Trump's staunchest critics, claimed in a Friday appearance on Good Morning America that Trump's act was necessary following Tuesday's chemical attack.
"President Trump should make his case in front of the American people and allow their elected representatives to debate the benefits and risks of further Middle East intervention to our national security interests, " Lee said in a statement.
Turkey, which also is involved in the fighting, has long pushed for Assad's ouster.