PowerofHoodoo has already answered, but I want to share how I have found the name of that piece before. The way I found the answer took a couple layers of investigation. Right after the part they play is a part that's incorporated into a tune in the video game "Yoshi's Cookie." If you research that tune, you'll find that's adapted from a piece called "Csikós Post" (this time around I found a comment on YouTube that said it; I don't recall how I found it out the first time). In the Wikipedia article about "Csikós Post," the second paragraph says, "The melodic line in which the piece transitions to C major is a variation on a melody from Franz Liszt's Hungarian Rhapsody No. 2."
Here are some titles of pieces that, though the titles may be unfamiliar, contain melodies you might recognize:
- "Korobeyniki" (Russian folk song)
- "Morning Mood" by Edvard Grieg
- "Entrance of the Gladiators" by Julius Fučík
- "Sabre Dance" by Aram Khachaturian
- "Soulful Strut" by Young-Holt Unlimited (an instrumental of a song called "Am I the Same Girl")
- "Pick Up the Pieces" by Average White Band
I think "Stars and Stripes Forever," "In the Mood," "Misirlou," and "Wipeout" are well known enough by both melody and title that they don't need to be on that list. I'm on the fence about "Sabre Dance." I want to include "Sunchyme" by Dario G, but I'm probably overestimating its importance because I just found out a few days ago that that was the title of that "Hey-a-ma-ma" song.