Halil Altintop's overhead kick brought Augsburg level after Raffael had given the Foals the lead.
And Tobias Werner then scored the winner 10 minutes from time to lift Markus Weinzierl's men above their hosts and Mainz, who play on Sunday, and into sixth place in the standings.
Gladbach, still seeking their first win of 2014, went in front only five minutes in when Raffael was allowed to close in on the penalty area without being challenged, and he took full advantage with a powerful shot into the left-hand corner.
Five minutes later, the Foals had the chance to double their lead when Filip Daems sent in a low cross from the left which was met by Max Kruse, but his shot was straight at Alexander Manninger.
A Raffael free-kick stung the palms of Manninger before the Brazilian failed to beat the Austrian in a one-on-one opportunity.
Their rivalry continued with the Augsburg goalkeeper again denying the Gladbach forward with another direct free-kick, and the hosts' failure to add a second came back to haunt them when Altintop levelled 10 minutes before the break.
Werner's cross was headed into the middle by Andre Hahn, fresh from his first call-up to the Germany squad, and the Turkish forward scored with an acrobatic effort with his back to goal.
Hahn and Werner both had chances to give Augsburg the lead in an evenly balanced second half which could have gone either way.
Patrick Herrmann wasted Gladbach's best chances before Werner added the finishing touch to Hahn's cross after a fine Augsburg counter-attack to earn the visitors all three points, taking them into the top six.