Jurgen Klopp's men were arguably the better side during the first half but still found themselves behind five minutes after the break when Raul Bobadilla pounced on a loose ball inside the area to slot home.
The visitors were reduced to 10 men 14 minutes later when Christoph Janker hauled down Pierre-Emerick Aubameyang with a professional foul on the edge of the area, allowing the hosts to increase the pressure.
However, the 2011 and '12 champions could not find a breakthrough, succumbing to their first defeat to Markus Weinzierl's men, much to the disdain of the home faithful, who booed their team off the field.
Dortmund remain bottom of the league, with an extraordinarily bad run seeing them win just four games in 19 this term.
In contrast, Augsburg are now on a run of three consecutive league victories, defying their pre-season critics to move into the UEFA Champions League spaces ahead of Bayer Leverkusen, who continued their own bid for a top-four finish with a 1-0 win away to relegation-threatened Hertha BSC.
Elsewhere at the bottom, Hamburg emphatically ended a barren run of four games without a goal with a crushing 3-0 win at fellow strugglers Paderborn.
A second-minute penalty from captain Rafael van der Vaart saw the visitors on their way, before second-half strikes from Marcell Jansen and Zoltan Stieber wrapped up the win, which gives Josef Zinnbauer's side a two-point cushion above the relegation zone.
Meanwhile, Paderborn have now recorded just one win in their past 11 Bundesliga outings.
Werder Bremen secured their third straight win with a 2-1 victory at Hoffenheim, while Cologne were held to a 0-0 draw at home to Stuttgart.
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