With central defender Mats Hummels making his comeback after more than two months out injured, Aubameyang helped take his team on to 36 points, one behind second-placed Bayer Leverkusen who host Stuttgart on Saturday.
The Gabon midfielder proved a worthy replacement on the right for Poland international Blaszczykowski who last week sustained a cruciate knee ligament injury that will keep him out for the rest of the season.
Aubameyang completed a fine move with a close-range header after 31 minutes.
Benjamin Kessel equalised for Braunschweig early in the second half before Aubameyang made it 2-1 following a quick break in the 65th minute, his 11th league goal of the campaign.
Dortmund, who take on Zenit St Petersburg in the Champions League last 16 next month, looked sharper than they have in recent weeks and missed several chances to add to their goal tally in a dramatic finale in which both sides hit the woodwork.
Leaders Bayern Munich, 13 points clear of Leverkusen, entertain Eintracht Frankfurt on Sunday.