US Secretary of State
The cautious statement came after Tillerson's first meeting with Russian counterpart Sergei Lavrov on the sidelines of a G20 gathering in the German city of Bonn.
"The United States will consider working with Russia when we can find areas of practical cooperation that will benefit the American people," Tillerson told reporters after the talks.
"Where we do not see eye to eye the United States will stand up for the interests and values of America and her allies."
For his part, Lavrov stressed the common ground between Washington and Moscow.
"We cannot solve all problems... but we have a mutual understanding that where our interests coincide, and there are many such spheres, we must move ahead," Lavrov said in comments televised in Russia.
Tillerson was in the global spotlight as he made his debut as America's top diplomat after President Donald Trump promised to put US interests first while also offering a softer line on Moscow.
But the closely watched encounter with Lavrov took place as Washington reels from the shock resignation of national security adviser Michael Flynn over contacts with Moscow's ambassador and allegations of Russian meddling in Trump's election last year.
For his part, Lavrov told Tillerson that Moscow does not meddle in other countries' internal affairs.
"You should know we do not interfere in the domestic matters of other countries," he told reporters.
But in Moscow, the Kremlin voiced impatience over the lack of progress in bolstering ties since Trump moved into the White House.
Kremlin spokesman Dmitri Peskov said the two countries were "wasting time," especially when neither on their own could solve pressing world problems.
Separately, President Vladimir Putin called for restoring links between US and Russian intelligence agencies, saying "even a simple exchange of information" could strengthen the fight against terrorism.
In an apparent rebuttal, US Defence Secretary James Mattis said at a NATO meeting in Brussels that Washington was not ready "right now" for military collaboration with Russia.
Both Tillerson and Mattis were making their diplomatic debuts in Europe, with their counterparts eager to find out what Trump's "America First" policy means for the rest of the world.
The US billionaire had alarmed allies by signalling he might reconsider sanctions against Russia and calling the NATO alliance into question, at a time when member states are nervous about a resurgent Moscow.
But both Mattis and Tillerson, who has rarely addressed the media since taking office, appeared to ease those concerns on their maiden European visits, signalling no major shift in policy.
British Foreign Secretary Boris Johnson, who also had a sitdown with Tillerson in Bonn, welcomed the measured stance.
"You have got to engage with Russia but you have got to engage in a very guarded way," he told the BBC.
"We don't want to get into a new Cold War. That's something London and Washington are completely at one on. But nor do we want Russian behaviour to continue as it is. Rex Tillerson has been very clear about that."
Ties between the West and Russia have plunged to a post-Cold War low over Moscow's 2014 annexation of Crimea and its support of pro-Moscow rebels in the Ukraine conflict.
Tillerson, a former ExxonMobil chief executive known for his close business ties to Russia, on Thursday urged Russia to adhere to the Minsk peace accords, following a flare-up in fighting in eastern Ukraine.
"As we search for new common ground we expect Russia to honour its commitment to the Minsk agreements and work to deescalate the violence in Ukraine," Tillerson said.
He made no mention however of Trump's campaign pledge to review US sanctions against Russia.
Lavrov said the issue had not been discussed.
The G20 gathering runs until Friday.
Host nation Germany billed the two-day meeting as a chance for the club of leading and developing economies to discuss how to work together on global challenges from climate change to the conflicts in Ukraine, Yemen and Syria.
The meeting serves as a prelude to the G20 summit in July, likely to be the first time Trump will meet Putin in person.