Menshikov had his debut role in the 1980 television film Zhdu i nadeyus (Russian: Жду и надеюсь; English: They Wait and They Hope) by Suren Shazbazyan. The next year he made his film debut in Family Relations, which was his first collaboration with Nikita Mikhalkov. Menshikov then went on to star a series of successful films that made him a star in the Soviet Union, including Flights in Dreams and Reality, My Favourite Clown, Through Main Street with an Orchestra and Moonzund. He garnered further critical acclaim for his performance in Dyuba-Dyuba (1992), for which he was nominated for a Nika Award.