Dortmund went ahead after 22 minutes through Julian Schieber but found themselves behind at the interval after goals from Daniel Baier and Kevin Vogt stunned the hosts.
But Schieber restored parity by netting his second goal in the 52nd minute before Neven Subotic put Dortmund ahead in the 64th minute and then Robert Lewandowski sealed the triumph in stoppage time.
The win moved Dortmund onto 55 points - 20 points behind Bayern but six ahead of third-placed Bayer Leverkusen - with six games left, while Augsburg remained in the relegation play-off spot on 24 points.
Dortmund coach Jurgen Klopp made seven changes to his starting XI after a 0-0 midweek draw in first leg of their Champions League quarter-final tie at Malaga,
The much-changed line-up took the lead in the 22nd minute. Moritz Leitner fed Schieber, who slotted past Augsburg goalkeeper Alexander Manninger from 12 yards.
The hosts went close to doubling their advantage three minutes later but Manninger made a fine save from Nuri Sahin.
And they were made to pay before the break. Augsburg equalised in the 43rd minute when Schieber's headed clearance from a corner fell only to Baier, who unleashed a first-time drive from just outside the area which flew past Dortmund keeper Mitchell Langerak.
Augsburg grabbed a second goal within 80 seconds when Tobias Werner's shot was deflected away by Langerak but right into the path of Vogt, who notched his first Bundesliga goal.
Klopp introduced Lewandowski and Mario Gotze in the 52nd minute, and the move paid almost instant dividends as Lewandowski supplied a through-ball to Jonas Hofmann, who crossed for Schieber to nick his second goal of the day to make it 2-2.
Augsburg nearly restored their lead in the 55th minute when Jan Moravek snatched the ball from Marcel Schmelzer and rounded Langerak, but he could not get enough power on his shot and Kevin Grosskreutz cleared off the line.
Dortmund moved up a gear in the final half-hour and went 3-2 ahead in the 64th minute, Subotic converting Gotze's free-kick from close range.
The visitors had one more chance to get back on level terms in the 86th minute, but Langerak made a fine save to prevent Dong Won Ji's shot from finding the top-right corner.
Dortmund wrapped up the points two minutes into stoppage time, Gotze doing well to retain possession and tee up Lewandowski, who fired home for the 10th consecutive Bundesliga match.