Die Fohlen had lost all three of their previous matches on their travels this term, but had picked up maximum points from their three games on home soil.
Hahn made an intelligent run to beat the offside trap and then tucked the ball away into the bottom corner from a tight angle, but Augsburg's advantage lasted for just six minutes.
The next half-hour of the match was largely contested in the middle of the park, but the Gladbach turnaround was complete when Kruse turned provider for substitute Branimir Hrgota on 72 minutes.
Three minutes after his introduction, the Swedish forward pulled off his marker to fire Kruse's cross beyond former Arsenal goalkeeper Alex Manninger on the half-volley.
However, the hosts snatched a late draw and limited Gladbach's first away success of the campaign to just a point, when Milik tapped in at the far post after Hahn's prod was parried by Marc-Andre ter Stegen.