But so far, it appears the two countries are growing further apart.
Mr Tillerson said there was "a low level of trust" between both countries and such a relationship could not be allowed to continue between the world's foremost nuclear powers.
In Washington, at a news conference with the North Atlantic Treaty Organisation secretary-general, President Trump was asked by the BBC's Jon Sopel if he thought it was possible that Syrian forces could have launched the alleged attack without Russian Federation knowing about it.
"I want to repeat my condolences over the terrorist attack to Saint Petersburg and reiterate that we should fight terrorism with strengthened global cooperation", noted the top diplomat of Iran during the meeting with his Russian counterpart.
"We have insisted that we have a very thorough investigation", Lavrov said at the news conference with Tillerson.
An official said the White House was "very concerned" about interference in the October elections, blaming them for an alleged coup attempt.
China - one of the five permanent Security Council members which hold veto power - abstained from the vote, as did Ethiopia and Kazakhstan.
In response to last week's chemical attack in the rebel-held town of Khan Sheikhoun, the USA fired 59 Tomahawk missiles against the Shayrat airfield on Friday, in the country's first strike against the regime since the war broke out six years ago. During his presidential election campaign, Trump had indicated that he would be less willing than some of his predecessors to conduct military operations against other countries.
This comes as the rift between the USA and Russian Federation deepens over the recent happenings in Syria.
The high-level talks in Moscow clearly failed to bridge the widening gap between the USA and Russian Federation on Syria.
"That could happen, and it may not happen", he said.
"Where is the proof that Syrian troops used chemical weapons?"
"I think it's certainly possible".
The newly hardened view of Moscow comes as the president has tried to shake suspicions about the motives behind his campaign calls for warmer relations.
"Putin is the leader of Russian Federation".
The West has accused Assad's forces of carrying out the attack in Idlib province that shocked the world.
Russia's United Nations envoy before the vote said the resolution served no objective and pointed out that the executive council of the Organization for the Prohibition of Chemical Weapons is due to meet in the Hague Thursday to discuss the incident, and urged all parties to cooperate with any independent worldwide investigation launched by that organization.
The dim view of U.S.
However, Trump also said Mr Tillerson's Moscow meetings went better than expected.
Western leaders backed the US action, saying Assad had brought it on himself. But Tillerson repeated the administration's new belief that "the reign of the Assad family is coming to an end".
Russian President Vladimir Putin also commented on the tensions between Washington and Moscow on Wednesday. "We have no firm information to indicate that there was any involvement by Russian forces into this attack", he said. "We're going to see how that all works out". "I do not know who saw them".
Most recently, Russian Federation blocked a council resolution in February condemning Syria for chemical attacks using chlorine gas.
Still, Tillerson sought to stress the positives from his meetings.
However, Lavrov added some progress had been made on Syria at the meeting, adding that a working group would be set up to examine the poor state of US-Russia ties.
There were only a handful of issues Tillerson and Lavrov said they broadly agreed on - that the Korean peninsula should be denuclearized, Syria should be "unified and stable" after the Islamic State is defeated, and that there should be more communication between USA and Russian diplomats and militaries.