Paul Verhaegh put the Bavarians in front from the penalty spot midway through the first half and two goals in three minutes from Andre Hahn effectively sealed victory for Markus Weinzierl's men.
Tobias Oehrl pulled one back against his former club early in the second half, but Halil Altintop completed the rout to move Augsburg onto 23 points with one game still to go before reaching the midway stage of the season.
Augsburg had already played their best ever first half of a Bundesliga season, yet they were clearly keen on adding to their haul of 20 points against bottom-placed Braunschweig.
It took them 23 minutes to break the deadlock when Norman Theuerkauf fouled Tobias Werner inside the penalty area, and Verhaegh converted from the spot.
The Bavarians were celebrating again just seven minutes later when Matthias Ostrzolek's cross took a deflection to take it over the luckless Theuerkauf and find Hahn, who turned the ball in from six yards out.
Hahn was celebrating yet again three minutes later after scoring the goal which effectively ended Braunschweig's hopes.
Werner set him up, although he still had a lot of work to do on the edge of the area as he dribbled his way past Marcel Correia before picking his spot in the bottom right-hand corner of Daniel Davari's goal.
Former Augsburg midfielder Oehrl reduced the arrears early in the second half, signalling Braunschweig's second-half intentions with a shot from outside the area which nestled in the top left-hand corner of Marwin Hitz's goal.
Dreams of a recovery did not last long, though, as Altintop restored the home side's three-goal advantage in the 76th minute, lobbing Davari in style to put the cherry on top of Augsburg's performance and seal them a win which puts them in the hunt for a top six finish.