Tillerson made the comments during the Association of Southeast Asian Nations' Regional Forum in Manila, Philippines, on Monday.
The Trump administration's top U.S. diplomat said he told Lavrov that "meddling in the elections is certainly a serious incident" during a bilateral meeting between the two Sunday.
"Trying to help Russia understand just how serious this incident had been and how seriously it had damaged the relationship between the U.S. and the American people and the Russian people," Tillerson said.
"I said this had created serious mistrust between our two countries and that we simply have to find some way to deal with that."
The meeting between Tillerson and Lavrov occurred four days after President Donald Trump signed into law a bill to strengthen sanctions against Russia, North Korea and Iran.
The bill, which the House and Senate passed by wide margins last week, toughens sanctions on the three nations -- specifically with Russia over its presence in Ukraine and Syria, as well as the U.S. intelligence community's assertion that Moscow tried to influence November's presidential election in Trump's favor.
After Congress approved the sanctions, but prior to Trump signing the bill, Russian President Vladimir Putin ordered the United States to reduce its diplomatic personnel in Russia by 755.
The Russian Foreign Ministry also said it would seize two U.S. diplomatic properties -- cottages just outside Moscow's city center and a warehouse facility in the capital.
Tillerson also said Monday that Trump's administration is preparing to respond to Moscow's demands.
Despite the U.S. sanctions and Russia's demands, Tillerson said cooperation with Russia is a necessity.
"We want to work with them on areas that are [of] serious national security interest to us, while at the same time having this extraordinary issue of mistrust that divides us," Tillerson added. "That's just what we in the diplomatic part of our relationship are required to do.
"I don't think it is useful to just cut everything off on one single issue."