The 28-year-old scored two quickfire goals within two minutes as BVB returned to the summit of the Bundesliga.
Maximilian Philipp opened scoring for the hosts two minutes into the encounter before Sokratis Papastathopoulos doubled the lead just on the stroke of half-time.
After the restart, Aubameyang scored his third league goal of the season from the penalty spot when the Video Assistant Referee spotted Cologne defender Lukas Klunter with a handball in the 59th minute.
A minute after his first goal, the Gabon international made it two after slotting Lukasz Piszczek's low cross home, to make it 4-0.
Philipp completed his brace and the rout with his match winning goal in the 69th as Peter Bosz’s men continue their impressive start to the season with no goals conceded in four league games.
Dortmund sit atop of the Bundesliga with 10 points from four games and will lock horns with Hamburger SV for their next league game on Wednesday.