The 24-year-old Gabon international has agreed a contract that will keep him with the Bundesliga outfit until 2018.
The Champions League finalists are thought to have beaten Tottenham Hotspur and Newcastle United to Aubameyang's signature.
Dortmund sporting director Michael Zorc told the club's official website: "We are very pleased that in Pierre-Emerick Aubameyang we were able to sign a player we really wanted.
"He is offensively very versatile, has a lot of pace in his game, is a threat in front of goal and will provide us with more options."
Aubameyang scored 19 goals for Saint-Etienne last season.