The home side went agonisingly close to opening the scoring with five minutes on the clock when Christoph Baumgartner’s effort from the edge of the area went just wide of the right post.
All in all, the midfielder had an inauspicious opening to the contest, as 18 minutes later, he conceded a penalty.
An effort from the edge of the area was deflected wide of the post by the Austria international, but referee Martin Petersen was instructed by the VAR to consult the pitchside monitor.
There, he saw that Baumgartner had handled the ball with an outstretched arm and therefore awarded a spot-kick, yet centre-back Kevin Schlotterbeck’s penalty was well saved low down by Maarten Vandervoordt.
Nevertheless, the visitors bounced back from that missed opportunity and deservedly took the lead six minutes before half-time.
A through ball played Fabian Rieder in, and from the left side of the area, he pulled it back to Robin Fellhauer, whose first-time finish beat Vandervoordt to give the Fuggerstadter the advantage.
Victory over Hamburg last week ended a three-game winless run for Leipzig in all competitions, yet they were struggling to build on that here and continued to be frustrated after the restart.
In the second period’s first opening, some last-ditch defending from the away side ended with the ball finding its way to Baumgartner, whose effort was tipped over the bar by Finn Dahmen. Ridle Baku was found unmarked from the resulting corner, but his header was straight at the visiting shot-stopper.
Augsburg had won just one Bundesliga away match in which they scored first this season, but they continued to ride their luck when Castello Lukeba headed onto the left post as another opportunity went begging for Die Roten Bullen.
An equaliser felt inevitable for the hosts, though, and with just under 15 minutes remaining, they eventually drew level when Yan Diomande forced his way into the area and fired beyond Dahmen to bring them level.
Augsburg very nearly snatched victory in the closing stages when Rodrigo Ribeiro dinked the ball over Vandevoordt, but his effort came back off the post.
They were made to pay for this missed opportunity in very unfortunate circumstances, as a cross into the area was pushed away by Dahmen before ricocheting off Chaves and into the net to spark wild scenes of celebration in Saxony.
The result sees Leipzig move up one place to fifth and level on points with fourth-placed Stuttgart, while a dejected Augsburg do at least remain ninth and still with a healthy nine-point buffer to the relegation zone as their three-game unbeaten run was ended in cruel fashion.
Flashscore Man of the Match: Yan Diomande (RB Leipzig)