Midnight in Moscow does come from a Russian song but, according to Wikipedia, it was composed in 1955.

"Moscow Nights" (Podmoskovnye Vechera) is a Russian song, one of those best known outside its homeland.

The song was originally created as "Leningradskie Vechera" ("Leningrad Nights") by composer Vasily Solovyov-Sedoi and poet Mikhail Matusovsky in 1955 but, at the request of the Soviet Ministry of Culture, "Podmoskovnye Vechera" was prepared.

The British jazz group, Kenny Ball and his Jazzmen, had a hit with the song in 1961 under the title "Midnight in Moscow". This version peaked at number two on the U.S. Billboard Hot 100 chart in early 1962; in March that year it spent three weeks at number one on the American Easy Listening chart.

In 1962, at the height of the folk revival in the United States, the song was recorded by The Chad Mitchell Trio on its popular live performance album "At The Bitter End" on Kapp Records.