Dortmund, who wasted a second-half penalty, bounced back from their midweek home loss to Real Madrid in the Champions league to move five points clear at the top of the Bundesliga on 19 points from seven matches.
Second-placed Hoffenheim are away to Freiburg on Sunday while champions Bayern Munich, who are third on 13, face Hertha Berlin.
The 28-year-old Kagawa scored his 38th goal in the league to surpass Shinzi Okazaki, with a superb chip over keeper Marwin Hitz.
Earlier, Augsburg's Caiuby had cancelled out Andriy Yarmolenko's fourth-minute lead for Dortmund.
The visitors should have added several more goals in a one-sided first half with Bundesliga top scorer Pierre-Emerick Aubameyang failing in his one-on-one against Hitz and Kagawa firing over the bar from close range.
"This is our worst game since I became coach," said Dortmund's Peter Bosz, who took over this season. "In the first half we had the spaces but played bad football. In the second half we played no football at all."
"Every team has a couple of such games every season so I am happy that we won this game."
Dortmund, undefeated in the league, inexplicably stepped off the gas after the break, allowing Augsburg a string of fine chances.
Keeper Roman Buerki came to the rescue repeatedly to protect their slim lead which Aubameyang again should have increased in the 79th.
He wasted his penalty, however, chipping the ball into Hitz's hands.
Hanover 96 suffered their first loss of the season after Borussia Moenchengladbach's Thorgan Hazard scored with a penalty deep in stoppage time for a 2-1 win that moved his team up to sixth on 11.
Sebastian Haller also scored a last-gasp winner with a spectacular sideways volley to give 10-man Eintracht Frankfurt their first home win by the same score against VfB Stuttgart.
Hamburg SV and Werder Bremen shared the points in their goalless northern derby as both teams battle to stay out of the relegation zone.
(Reporting by Karolos Grohmann; Editing by Clare Fallon)
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