Benjamin Kessel cancelled out Aubameyang's opener to briefly put Braunschweig on course for an unlikely point, before the Gabon striker netted his second to earn Jurgen Klopp's side a first win in five matches.
Last year's Champions League finalists soon found their feet, though, and smart interplay between Robert Lewandowski and Aubameyang led to the latter heading home from close range on 32 minutes.
The same pair combined moments later, but the offside flag was raised as Dortmund's goalscorer fired into the bottom corner.
He was not to be denied a second after the interval, although it was Braunschweig, who have only won three league games so far this season, that found the net next to surprise Dortmund.
However, Braunschweig's parity lasted only 11 minutes as Aubameyang shook off the attention of Kessel to collect Marco Reus's pass and drill a shot through Davari to earn Dortmund their first win of 2014.