Or did Doom come before that?
EDIT: Never mind. Marathon '94, Doom '93 and Wolfenstein 3D '92
What about the real origins?
There are two.
http://en.wikipedia.org/wiki/Spasim from http://en.wikipedia.org/wiki/Maze_War
I've heard of a game about running from a T-Rex that was done before GRAPHICS (it was ASCII) that was in first-person as well, but I can't find it or remember the name.
Still, I think going almost 10 years before ME gives you the picture (Monster Maze is my age, and was also first-person).
EDIT: Original Team Fortress for Quake is the earliest class-based FPS game I know of, and it was the first to make headshots actually work properly - but not the first to attempt it)