Yes, it is Khachaturian's "Sabre Dance". He was an Armenian composer, hence also a Soviet composer, so perhaps that was one of the reasons for the choice of his music in a film satirizing Communism (among other things, such as the way Wilder, an Austrian, makes fun of the Germans). But, in line with the frenetic pace of that piece of music, the film does not really satirize the "evils" of Communism, but rather the utter and zany craziness of such a stupid system.
You haven't really understood the movie, have you? It mocks all systems alike, making fun of all protagonists (Piffl for communism, MacNamara for capitalism and Schlemmer for militarism).
It's the "Sabre Dance" from Gayaneh by Aram Katchaturian. Previn did the music for this film, so he may have altered it some. "Flight of the Bumblebee" is by Rimsky-Korsakov.