Putin again denied trying to influence the election, which top U.S. intelligence agencies have concluded he personally ordered. Tillerson called Moscow's attempts to affect the election a "hindrance in the ability of us to move the U.S.-Russia relationship forward."
Trump, however, wanted to talk about how the countries "go forward" rather than "spending a lot of time having a disagreement" over election interference, Tillerson added. "Everybody knows we have a disagreement," he said.
Lavrov told reporters that Trump accepted Putin's assurances that Russia did not interfere in the election, though the White House later disputed that to NBC News. Trump has repeatedly muddied the waters over whether Russia tried to affect the election, despite the top intelligence agencies findings.
Tillerson said the U.S. is looking to seek assurances that Russia will not meddle in American affairs. He highlighted a Russia sanctions bill that passed the Senate 98-2.
Before the meeting, Trump said he felt his relationship with Putin was "going very well."
"We're going to have a talk now, and obviously that will continue. But we look forward to a lot of very positive things happening for Russia, for the United States, and for everybody concerned," Trump said.
He added that it was an "honor" to be with Putin. The Russian president said he was "delighted" to meet Trump "personally" before the pair shook hands and the throng of reporters and photographers left. Trump did not answer shouted questions about whether he would confront Putin about Russian efforts to interfere in the 2016 election.
Trump came into office seeking a reset of relations with Russia and hoping to cooperate in some policy areas, like fighting the terror group ISIS. However, Putin's support for ruthless Syrian President Bashar Assad, among other issues, has emerged as a hurdle to their working together.
Tillerson said Friday that he sees no long-term role for the Assad family in Syria going forward. How Bashar Assad "leaves is yet to be determined," he said.
On Thursday, Trump criticized Moscow's actions in the strongest terms he has used as president and hinted at possible topics the pair could address.
"We urge Russia to cease its destabilizing activities in Ukraine and elsewhere, and its support for hostile regimes including Syria and Iran, and to instead join the community of responsible nations in our fight against common enemies and in defense of civilization itself," Trump said in Warsaw.
But just hours before the speech, Trump made a statement that appeared to please the Kremlin. He acknowledged that Russia may have interfered in the 2016 U.S. election but said that other countries may have meddled, too.
"Nobody really knows. Nobody really knows for sure," Trump said.
— The Associated Press contributed to this report