Perhaps the reason why this episode reminds you of The Twilight Zone is because the music used in the CBSRMT *is* from The Twilight Zone TV series.  The network owned the scores used in the series so it was logical for it to provide them as the basis for the canned music used for the radio shows.  Some of the best (IMHO) cues were written by Bernard Herrmann, the legendary film composer who wrote spooky film scores such as Psycho and The Day the Earth Stood Still, and can be found in the first season episodes "The Lonely", "Where is Everybody",  and (I think) "Judgment Night".