It's usually the other way around for me. Instead of "feeling" what I compose, I compose the feeling I want to convey. But the feeling comes more from music appreciation. Depending on the composer, I can usually feel what the composer's message is. I think a good example of this is Tchaikovsky's last movement of his last symphony. Or lets go ancient! Baroque has emotion too, such as Bach's Toccata in D minor. Here, we have music from the 1700's and Bach was able to create a story out of this piece. Absolutely wonderful because I feel this specific piece is very freestyle, and almost a little like Chopin's Ballades (in terms of storytelling). And speaking of Chopin, he was also very good at creating emotion using the piano, as any one of his Ballades is literally equivalent to a short story in literature.
Going contemporary, Schindler's List comes to mind as very, very emotional and something I think most people can feel. It conveys a slightly different message than Tchaikovsky's symphony 6 (*without knowing anything about this movie...it sounds more like a love story gone bad, or someone who is dealing with a great loss). This piece is musically less sophisticated than Tchaikovsky's 6th, but more modern and may appeal to a broader audience. And in terms of Tchaikovsky's 6th, he's conveying death and darkness of some type in a very eloquent manner. Both works however do the job of getting the listener to feel sadness though, and I think even an inexperienced listener can understand that the last movement of Tchaikovsky's 6th is absolutely devastating.
* Peter told me what this movie was about. After research, it appears to be about a man named Oskar Schindler, who became a humanitarian and gave up his savings to help those in need during WWII. I had no idea. No wonder this music is so emotional.