Janis Joplin covered a lot of songs in her day, this original version of Cry Baby sounds a lot more Gospel than her blues.
In fact, this song was written by two American born sons of Eastern European Jews. Bert Berns also wrote Hang on Sloopy, Twist and Shout and many others. He died five years after this song came out, before Janis took it back up the charts. Jerry Ragovoy wrote Piece of My Heart, Get it While you Can (covered by Janis as well) and Time is on My Side among many others. He died earlier this year.
The song is written in 6/8 time, so it sounds like a fast waltz. The drummer uses almost exclusively snare, pushing the beat along, never getting in the way of everything else going on in the song. Lyrically it's a man singing to a woman who loves another man. The singer presumes that the other man will do her wrong, and she will come running to him. It clearly belongs in lots of movies. I think if Bruno Mars did a cover of it and it was put in a movie starring Jonah Hill, Kristen Bell and Topher Grace it would climb the charts again.
There's a bit of twangy guitar through the song, which makes it sadder. The piano gives us most of the back bone of the song. You need a versatile instrument that can go all the way up and down half a dozen octaves just to try to keep up with the vocals. Garnet Mimms sings down into the lower part of the baritone range and all the way up into the high tenor. The background group, The Enchanters help keep him aloft up there in the stratosphere. Jerry and Bert also produced the track, making sure that the brass section that comes in at 1:50 adds just enough sad, and then fades out. Garnet does a kind of spoken word thing that was very popular in R&B/do-wop style hits of the early 60s around 2:20. The strings start to swell and the song lifts us up into the end, with a sax choir responding to the vocalists appeal as we fade out.