Hahn, who earned a maiden call-up to the Germany side for the friendly with Chile earlier this month, will link up with Gladbach in June after penning a four-year deal with Lucien Favre's side.
The 23-year-old becomes the second player to agree to join Gladbach next season, following Basel goalkeeper Yann Sommer, who will replace the departing Marc-Andre ter Stegen.
Former Hamburg man Hahn has enjoyed an excellent season for Augsburg so far, scoring 10 goals in 27 Bundesliga appearances.
And Gladbach sporting director Max Eberl revealed that they beat a number of other suitors to Hahn's signature.
"Andre Hahn has undergone an incredible development at Augsburg this season," Eberl told Gladbach's official website.
"He is one of the top-scoring wingers in the league and will certainly strengthen us in attack.
"We had strong competition in our efforts to sign him and are pleased that he has chosen us."
If Hahn can remain in Joachim Low's squad for Brazil 2014, it would cap an impressive 18 months for the winger, who only joined Augsburg from third-tier outfit Kickers Offenbach in January 2013.
