Klein leaves Brisbane after scoring nine goals across 45 A-League appearances.
The midfielder joins fellow Australians and Socceroos mainstays Jackson Irvine and Connor Metcalfe at the Hamburg-based club.
"I'm really looking forward to joining St Pauli and getting to know my new teammates, a different country and a new league," Klein said in a statement.
"The club has successfully integrated and developed international players especially from Australia on several occasions in recent years.
"The club's management have outlined an exciting prospect for me, which is why I'm convinced this is the right move for me."
Brisbane said they secured "a significant transfer fee and a sell-on agreement" as part of the deal for the Gympie-born midfielder.
"Nothing has been given to Sam, he has had to work hard for every professional opportunity, and we couldn't be happier to see one of our own progress to a top five league globally," said Roar chief operating officer Zac Anderson.
"This deal is important, as it's another piece of evidence that Brisbane Roar is a genuine springboard for young Queensland and Australian talent to reach the top of the world game."
St Pauli, a club known for its progressive stance on political issues, remain locked in a battle to avoid relegation from the Bundesliga.
The German side sit 16th with three games left to play and are currently on course to enter the relegation play-offs.
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