But the two nations remain at odds over USA airstrikes on Syria - following last week's chemical attack on Syrian civilians.
The United States said it attacked the airfield based on information it attained that the forces of Syrian President Bashar Assad launched a Su-22 fighter jet on April 4 from that base armed with sarin gas.
Earlier in an interview broadcast on Wednesday before the meeting with Tillerson, Putin said that mutual trust between Russian Federation and the US, especially on the military level, had eroded in the first few months of Trump's presidency.
Syria's government has denied responsibility for the gas attack in a rebel-held area of northern Syria that killed at least 87 people, many of them children.
His comments came moments before President Donald Trump said relations with Russian Federation had reached a "all-time low".
Russia's alleged meddling in the USA presidential election also hovered over the first face-to-face encounter between Mr Putin and a Trump administration Cabinet member.
But he said it was an open question where relations go from here.
He said working groups would be established to improve US-Russian ties and identify problems.
Mr Lavrov also said President Vladimir Putin is ready to return to implementing the memorandum of understanding on flight safety in Syria with the United States.
"Russia is open to this, open to dialogue with the U.S. in all different areas, not only to dialogue but to joint action aimed at results in the areas where this corresponds to the interests of both countries", said Lavrov. The diplomats know each other well from Tillerson's days as Exxon Mobil CEO. Mr Putin had even honoured Mr Tillerson with a friendship award.
Only weeks ago, it appeared that Trump, who praised Putin throughout the US election campaign, was poised for a potentially historic rapprochement with Russian Federation.
On the eve of the talks, far from trying to calm tempers, both sides escalated their rhetoric as the USA tried to prise Moscow and Damascus apart.
Russia's Safronkov called the USA strikes on a Syrian air base in retaliation for the Khan Sheikhoun a provocation, but Delattre said they were "a legitimate response to a mass crime which could not go unpunished".
"We do think it's important that Assad's departure is done in an orderly way", he continued.
Russian President Vladimir Putin said ties with the United States have deteriorated under the administration of Donald Trump as their top diplomats locked horns today over the Syrian conflict. Trump himself accused Putin of backing "an animal" and "a butcher", someone who is "truly an evil person".
MARGARET WARNER: In turn, President Trump had harsh new words for the Russians and their Syrian ally Assad in an interview that aired on FOX Business Network. According to Reuters, the Russian president said his country's relationship with the U.S. has "deteriorated" under Trump, "especially on the military level". How that occurs, we leave that to the process going forward. "If you have any examples I'd be glad if you could share them with me". "I like to think they didn't know".