My favorite Russian (Soviet) cartoon--and the one all my U.S. students love without exception--is Cheburashka. Cheburashka is a cute little creature of unknown origin who arrives to Moscow in a crate of oranges. Here is Cheburashka on the entry page of a wonderful Russian site devoted to this cartoon.
Based on a story by Eduard Uspensky, the Cheburashka cartoons are funny and even, for 1972, a bit subversive. Cheburashka meets his best friend Crocodile Gena when the crocodile posts personal ads for new friends. Cheburashka responds and they become fast friends. When they want to join the Pioneers (required communist scouting organization during the Soviet era) they are rejected because they don't know how to build a birdcage. Gena and Cheburashka triumph when they build a clubhouse and the snotty pioneers come around to their side.
There is a great villain--an old lady named Shapocliak who likes to play petty tricks on strangers and leads a pet rat on a leash--and wonderful music, including К сожалению, день рождения... ("Unfortunately, a birthday..." 8th song on the list), which serves as a Happy Birthday song in some circles. (There is no traditional Russian birthday song.)
The Cheburashka DVD has subtitles and is well worth it for children who can read. A great cartoon. And, it's really popular in Japan.