Marco Reus put Dortmund ahead in the 16th minute of a first half Dortmund largely dominated before being second best to Augsburg for much of the second half.
Second-placed Dortmund, 19 points behind runaway leaders Bayern Munich, could ill afford to drop points in a tight race for Champions League places.
With 41 points from 24 matches, they are one point ahead of Schalke 04, two in front of Eintracht Frankfurt and three clear of Bayer Leverkusen and RB Leipzig with 10 matches to go. Bayern were held 0-0 at home by Hertha Berlin on Saturday.
"We're disappointed with the performance today. We had a great chance to put some distance between us and the other sides," said Dortmund's German international forward Andre Schuerrle.
A cross from Schuerrle on the right to Michy Batshuayi at the far post was cut out by defender Martin Hinteregger only for Reus, who was following up, to steer the loose ball into the net.
Augsburg's equaliser to end a run of two defeats was no surprise after their better start to the second half.
Austria defender Danso tapped home the rebound from two metres after Swiss keeper Roman Buerki had parried his header at a corner.
Batshuayi, on loan from Chelsea, and Mahmoud Dahoud both missed chances to snatch a winner for the home side in the final minutes.
(Reporting by Rex Gowar, editing by Pritha Sarkar)
