I'm so glad to see a #russiabynatives post made by someone else. :) Thank you for forcing me to watch through it: it was an interesting story to find out about.
Here's the song. It's called "Подмосковные вечера" ("Evenings in Submoscow", Submoscow being the unofficial but very common name for the Moscow Oblast). There are many arrangements of it; this one is by Muslim Magomayev.
Sasha starts with another popular Russian song, "Очи чёрные" ("Black Eyes"). It might be worth a listen as well.