Menshov, who won the Oscar for best foreign film in 1981, died as a result of complications from Covid-19, Moscow film studios, Mosfilm, confirmed in a statement.
"We knew he was suffering from Covid-19, but in a mild form. It is absolutely horrible and unexpected," film director Vladimir Khotinenko told AFP.
"His death leaves a huge vacuum in our shared cultural space," Khotinenko added.
Moscow mayor Sergei Sobyanin said Menshov's death represented "a huge loss for our cinema and our culture".
Kremlin spokesman Dmitry Peskov said Russian President Vladimir Putin "expresses his deepest condolences".
Menshov was born in 1939 in Baku, then the capital of Soviet Azerbaijan.
He worked as an actor and then a director and gained international fame for his film "Moscow does not believe in tears" which won the Oscar for best foreign film in 1981.
It was only one of two Soviet films to win an Oscar -- the other being "War and Peace".
Menshov was beloved in Russia for his comedy "Love and Pigeons", released in 1984 and still one of the most watched films on Russian television.
Menshov, who directed 10 films, taught directing at VGIK.
