The U.S. and Britain say there is little doubt Assad's forces are culpable.
"Russia has really aligned itself with the Assad regime, the Iranians, and Hezbollah", Tillerson said before leaving Italy. "The facts are on our side", White House spokesman Sean Spicer told reporters.
"Based on these considerations, on April 6, the president directed military action consistent with our vital national interests to deter the use of chemical weapons", he said. US President Donald Trump will attend North Atlantic Treaty Organisation and G7 leaders' summits in late May - in effect, the diplomatic can has been kicked down the road until then.
Tillerson urged Russian Federation to rethink its alignment with the Assad regime, the Iranians, and Hezballah. I was trying to draw a distinction of the tactic of using airplanes to drop chemical weapons on population centres.
Lavrov said Russian Federation thinks there is no need for a U.N. Security Council resolution on the attack until the U.N. chemical weapons watchdog has completed an objective probe.
Officials said Tuesday that Russians were located on the small base before the chemical attack was launched.
Before meeting Putin, Tillerson received a chilly reception from Lavrov, who told Tillerson that Russian Federation was trying to understand the "real intentions" of the Trump administration.
However President Putin showed little inclination to respond to the USA initiative, saying he was aware of planned "provocations" to blame Syria's government for using chemical weapons.
In a four-page document, the White House sought to rebut many of Moscow's claims about the circumstances of the attack.
On domestic policy, Trump said he was making a second attempt to push through a broad health care overhaul before he turns to a sweeping tax rewrite, leaving the fate of both initiatives uncertain. USA officials in Washington have also raised that prospect.
Washington is leading a campaign of air strikes in Syria against Islamic State fighters and has backed rebels fighting against Assad during a six-year civil war, but until last week the United States had avoided directly targeting the Syrian government.More news: Minister: Iraq to boost crude oil production by year's end
Tillerson earlier met foreign ministers from the Group of Seven advanced economies and Middle Eastern allies in Italy. Now they call him a "serious" leader.
U.S. Secretary of State Rex Tillerson says he aims to clear up "sharp differences" with Russia as he opens a tense meeting with Russian Foreign Minister Sergey Lavrov.
It is not yet clear whether last week's strike will be a one off - a warning to Assad not to use chemical weapons against his own people again - or whether it signals a broader strategic shift in the USA approach to the civil war there, which is now in its sixth year.
Trump's spokesman Sean Spicer said Tillerson would go into the meeting with Lavrov to "make sure we let Russian Federation know that they need to live up to the obligations it has made" to halt Assad's chemical weapons use.
The incident which killed 87 civilians, including 31 children in the rebel-held town of Khan Sheikhun has been blamed on Syrian President Bashar Assad's regime by Turkey and the United States.
Only two weeks ago, Secretary of State Rex Tillerson and USA ambassador to the UN Nikki Haley hinted that the removal of Assad was no longer a U.S. priority.
The men know each other well from Tillerson's days as Exxon Mobil CEO. Putin had even granted Tillerson with a friendship honor.
"Putin knows that Trump personally degraded US intelligence credibility by attacking it over the Russian hacking and essentially going to war with the CIA and NSA", said Malcolm Nance, a veteran intelligence officer.
He said he had kept Comey in his job "because I want to give everybody a good, fair chance".
Assad was winning the war.
Allegations of collusion between Russian officials and Trump campaign associates also have weakened Trump's ability to sweeten any offer for greater cooperation, such as by easing economic sanctions on Moscow related to its 2014 annexation of Ukarine's Crimea region and support for pro-Russian separatists in eastern Ukraine.