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.
"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.
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.
Lavrov told Tillerson he "should know we do not interfere in the domestic matters of other countries."
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," 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.
US Defence Secretary James Mattis however 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 NATO into question, at a time when member states are nervous about a resurgent Moscow.
But both Tillerson, who has rarely addressed the media since taking office, and Mattis appeared to ease those concerns on their maiden European visits, signalling no major shift in policy.
British Foreign Secretary Boris Johnson 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.
He made no mention of sanctions and Lavrov said the issue had not been discussed.
Tillerson also stuck to conventional foreign policy in discussions with Japan and South Korea that condemned North Korea's latest ballistic missile launch.
"The United States remains steadfast in its defence commitments to its allies, the Republic of Korea and Japan, including the commitment to provide extended deterrence, backed by the full range of its nuclear and conventional defence capabilities," Tillerson said in a joint statement with the two foreign ministers.
The G20 gathering of leading and developing economies runs until Friday.
Trump is expected to meet Putin in person for the first time at a G20 summit in Hamburg in July.