Shinji Kagawa scored the winning goal as Borussia Dortmund continued their fine start to the season to stay top of the Bundesliga.
Andriy Yarmolenko, signed in the summer from Dynamo Kiev for 25m euros, opened the scoring after just four minutes before Caiuby equalised for Augsburg.
Former Manchester United midfielder Kagawa hit the winner midway through the opening half.
England Under-17 forward Jadon Sancho was an unused substitute for Dortmund.
He joined the club from Manchester City for a fee in the region of £10m in the summer.
Elsewhere in the Bundesliga, Eintracht Frankfurt beat Stuttgart 2-1, Wolfsburg drew 1-1 with Mainz and Borussia Monchengladbach were 2-1 winners against Hannover.
Source: BBC Sport