Promoted Leipzig are in second place on 49 points but missed the chance to close the gap to champions Bayern Munich who will move seven points ahead if they beat Cologne on Saturday.
Leipzig put the hosts on the back foot from the start but it was Augsburg who struck first when Konstantinos Stafylidis's powerful shot left keeper Peter Gulacsi with no chance.
Timo Werner levelled with his 14th goal of the season six minutes later and Marwin Compper headed in a corner at the far post in the 52nd to put Leipzig ahead.
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Augsburg, however, executed a textbook counter-attack with Martin Hinteregger beating Gulacsi on the hour. Stafylidis also hit the post late in the game as Augsburg climbed to 12th place in the table on 28 points.
(Reporting by Karolos Grohmann, editing by Ed Osmond)
