The club's director of sport Max Eberl announced the transfer on Sunday, shortly after his side's 3-1 win over Hamburg.
"We reached an agreement with Augsburg before this weekend, but we all agreed to make it public after the weekend's matches." Eberl told Sky television.
"We're very, very pleased that Andre has chosen us.
"The lad's career has taken on an excellent development. He's a very versatile player who embodies all that we want from a forward.
"He is going to sign a four-year contract."
Hahn has made a fairytale rise from the third division of German football to earning his first call-up to the Germany squad within the space of 14 months.
The 23-year-old was on the verge of hanging up his boots and pursuing a different career only three years ago before he was given a break in the third division with Koblenz, who soon after went bankrupt.
He moved on to Kickers Offenbach, where he caught the eye of Augsburg's director of sport Stefan Reuter in January last year.
He made his Bundesliga debut on January 18, 2013, and just over a year later, he received a phone call from Germany's assistant coach Hans-Dieter Flick calling him up for the friendly against Chile earlier this month.
Hahn, who thought somebody was playing a prank on him when Flick called, is now a candidate for Joachim Low's World Cup squad, while he has earned a move to a side who could offer him European football next season.