“I have now finished a conversation with Russian Foreign Minister Sergey Lavrov, who called to congratulate me on taking office,” Mr Cohen wrote in a tweet.
“We discussed a series of bilateral and regional issues, and the relations between the countries.”
He did not provide further detail about the call. Israel and Russia have been in discussions about a free-trade agreement since 2019.
The former Israeli government of Yair Lapid had faced calls to do more for Ukraine, including sending defensive weapons after Ukrainian President Volodymyr Zelenskyy asked Israel for its Iron Dome defensive missile system in October. The Iron Dome is widely seen as one of the world's most effective missile defence systems.
Some observers of Russia's war in Ukraine have argued that Israel would benefit from sending military equipment to help Kyiv fend off relentless Russian drone and missile attacks. Russia has received hundreds of Iranian so-called “kamikaze drones,” which could otherwise be used to bolster Iran's strength in the Middle East.
Benny Gantz, who was defence minister at the time, said that such military aid would not be forthcoming, although Israel sent some systems designed to detect incoming drones.
Mr Cohen, a senior member of returning Prime Minister Benjamin Netanyahu's new government, spoke to Mr Lavrov a day after he gave his first speech as Foreign Minister.
Mr Cohen said Israeli aid to Ukraine would continue, during his address on Monday, adding that there would be “less talk” on the Russia-Ukraine issue.
A potential warming of Israeli-Russian relations has seen backlash in the West.
US Senator Lindsay Graham said it was “unnerving” for Israel to speak less of the Russian invasion of Ukraine.
“I hope Mr Cohen understands that when he speaks to Russia's Lavrov, he's speaking to a representative of a war criminal regime that commits war crimes on an industrial scale every day,” Mr Graham wrote on Monday after Mr Cohen's announcement of the scheduled call.
During his brief stint out of office, Mr Netanyahu had called on the Russian President to end the war in Ukraine.
“We all have sympathy to Ukraine, it’s not even a question, and I’m no different,” he said during an interview with USA Today in October.