The West has accused Bashar al-Assad's forces of carrying out the attack in Idlib province that killed 87 people and shocked the world.
"Right now, we're not getting along with Russian Federation at all, we may be at an all-time low", Trump said.
"In general, primitiveness and loutishness are very characteristic of the current rhetoric coming out of Washington", Deputy Foreign Minister Sergei Ryabkov told Russia's state-owned RIA news agency.
Combination of file photos show Russian President Vladimir Putin (L) at the Kremlin in Moscow, Russia, Dec. 29, 2016 and U.S. President DonaldTrump (R) at a press conference at the White House in Washington D.C., the United States, on February 15, 2017.
Trump had frequently called during the 2016 U.S. presidential election campaign for warmer ties with Putin, despite criticism from lawmakers in his own Republican Party.
Lavrov said Moscow was hoping to understand Washington's "real intentions" and warned that Moscow considered it "fundamentally important" to prevent more "unlawful" U.S. strikes against its ally Syria.
Ten countries voted in favor of the resolution, two voted against it, and three countries abstained.
Russian president Vladimir Putin earlier said the two nations' relationship had deteriorated.
Lavrov said after the meeting that both sides agreed that there should be an investigation into the alleged April 4 sarin gas attack on the northern town of Khan Sheikhoun - a strike that killed almost 100 civilians, injured hundreds more and elicited a fierce USA retaliatory strike two days later.
The Trump administration concluded that Assad was behind the act, and on Thursday ordered a cruise missile strike against a Syrian airbase as a "proportional response".
Moscow's hostility to Trump administration figures is a sharp change from past year, when Mr Putin hailed Mr Trump as a strong figure and Russian state television was consistently full of effusive praise for him.
Moscow has insisted that the civilians who died there were killed by toxic agents released from a rebel chemical arsenal struck by Syrian warplanes. It denied preferring Trump to his Democrat rival, Hillary Clinton, during the election campaign and said it would judge his administration by its actions rather than its words.
Trump advisers: US seeks to fight IS and oust Syria's Assad
The reported investigation follows Moscow's condemnation of the U.S. missile strike on the Syrian airfield. She said late last month that "our priority is no longer to sit there and focus on getting Assad out".
Russian Federation blocked a Western effort at the UN Security Council on Wednesday to condemn the gas attack and push Assad to cooperate with global inquiries into the incident.
It was the eighth time during Syria's civil war that Moscow has used its veto power on the Security Council to shield Assad's government.
The Secretary of State described Russia's alleged involvement in the 2016 vote as a "serious issue" and "one that we know" about.
During his visit - the first to Moscow by a senior Trump administration official - Tillerson was expected to challenge Russian Federation to distance itself from Assad and his Iranian backers, an idea that the Kremlin dismissed as "absurd". Russia wants the U.S.to eliminate sanctions on Moscow related to its 2014 annexation of Ukraine's Crimea region and support for pro-Russian separatists in eastern Ukraine.
The US now assesses that Syria has re-established a unit of personnel associated with chemical weapons that existed before the 2013 agreement in which the Syrian government pledged to give up its weapons inventory.
"Russian Federation said no to cooperation with the UN's independent investigation", Haley said. "I suggest you look at this vote very carefully and heed this warning", she added.
President Donald Trump says Russian President Vladimir Putin is backing an "evil person" in Syria.
A fact-finding mission from the Organisation for the Prohibition of Chemical Weapons (OPCW) is investigating the attack.
At the start of his meeting with Lavrov, Tillerson said he wanted to "clarify areas of common objectives, areas of common interest - even where our tactical approaches may be different - and further clarify areas of sharp difference".
Tillerson reiterated that Assad was behind the attack.
"It'd be a fantastic thing if we got along with Putin and if we got along with Russia", Trump said.