Raul Bobadilla made the difference in the second half and the away side held on to the lead despite Christoph Janker being sent off
Dortmund, who have finished in the top two in the last four Bundesliga campaigns, have struggled this season amidst several key departures and serious injuries.
Terrible away form saw them trapped in the relegation zone as 2014 ended, but many hoped - and expected - that 2015 would be different for Jurgen Klopp's men.
But, after falling to the foot of the table on Saturday, a poor home defeat is sure to increase the pressure on Klopp.
Dortmund shaded the first half but found themselves behind when Raul Bobadilla pounced on a loose ball to smartly score in the 50th minute.
Augsburg defender Christoph Janker was given a straight red card for bringing down Pierre-Emerick Aubameyang midway through the second half and Dortmund upped the pressure as a result.
They were unable to find the all-important breakthrough, though, as Augsburg continued their outstanding season and climbed to fourth.
Dortmund were criticised for their lack of attacking verve in Saturday's goalless draw at Bayer Leverkusen and coach Jurgen Klopp bolstered his forward options with the inclusion of Aubameyang.
The Gabon forward returned from African Cup of Nations duty, while Ilkay Gundogan recovered from a thigh problem.
Dortmund were bright in the early stages, but had little to show for until they produced three efforts within 90 seconds.
Kevin Kampl initially fired wide in the 15th minute, before Marco Reus blasted into the side netting after an excellent first touch, and then Aubameyang toe-poked an effort at Augsburg goalkeeper Alex Manninger as the hosts turned the screw.
Aubameyang then found the back of the net, but his effort from one yard - which saw him bundle Reus' fierce effort in - was correctly ruled out for offside in the 26th minute.
After soaking up the pressure, Augsburg came agonisingly close to the opener, as, following a storming run, right-back Paul Verhaegh saw his low effort go narrowly wide, but neither side could break the deadlock in the first half.
Augsburg did get their goal, though, as Bobadilla produced a first-time finish five minutes into the second half.
Halil Altintop created havoc with a mazy run into the penalty area and even though he lost the ball, Dortmund panicked.
Marcel Schmelzer's tackle did enough to see off Ji Dong-won, but all it did was move the ball in Bobadilla's direction, with the Argentinian smashing past the onrushing Roman Weidenfeller and into the bottom-left corner from 12 yards.
Dortmund received a lifeline in the 63rd minute when Janker was given his marching orders and it ramped up the atmosphere inside Signal-Iduna Park.
Nuri Sahin fired the resultant free-kick well over, before Gundogan shot tamely at Manninger from 20 yards, but Dortmund were beginning to control the contest.
Immobile shot at Manninger in the 80th minute, and the goalkeeper was kept busy as Henrikh Mkhitaryan's half-volley required an excellent save.
In stoppage time, Immobile, inside the six-yard box, could only head straight at Manninger, and Schmelzer - next to try his luck - was also denied by the ex-Arsenal man.
Dortmund eventually ran out of time, though, and tensions were evident after the final whistle when captain Mats Hummels and Weidenfeller went over to speak with angry fans demanding answers for the club's pitiful form.