Well, Verklärte Nacht isn't atonal. Check out Pierrot Lunaire, for example:
Arnold Schoenberg - Catalogue of Works - Compositions - op. 21
If you can't play this from the site, there's also recordings on YouTube. (I believe you'll also find recordings of all the other pieces I posted here on this site.)
Or here, the six little piano pieces:
YouTube - 6 pieces for piano Op. 19 by Schoenberg (Beroff)
Or here something out of the 5 orchestra pieces Op. 16
YouTube - Arnold Schoenberg: Orchesterstuecke op. 16 1 of 2
Or the "Buch der Hängenden Gärten":
Arnold Schoenberg - Catalogue of Works - Compositions - op. 15
All of these predate the dodecaphonic phase of course, but they are called "atonal". (Schönberg himself didn't like that term being applied to his music. He preferred "atonical".)
Out of the works after the "invention" of dodecaphony, "Moses and Aron" is largely non-dodecaphonic, I believe:
YouTube - Arnold Schoenberg: Moses und Aron (Excerpts, 1 of 2)