Well, probably his best known song "Mack the Knife" comes from "The Threepenny Opera" by Kurt Weill, although personally I wouldn't count Weill as a well known composer. http://www.nodanw.com/biographies/kurt_weill.htm
I'm only guessing, because I don't know if he had a hit with it - but was it 'Stranger in Paradise'? That song is from the musical 'Kismet' which incorporated the music of Borodin. The tune of 'Stranger in Paradise' is part of the 'Polovtsian (spelling?) Dances' from the opera 'Prince Igor'