The hosts found themselves under siege in the first half but took the lead in the 48th minute through Konstantinos Mitroglou, who latched on to Luisao's headed flick-on from a corner and prodded home from close range.
Despite being second-best in most departments throughout most of the game, Benfica defended stoutly and also benefited from several strokes of luck as the Dortmund attackers were not able to produce the goods on the night.
Aubameyang was guilty of missing no fewer than three gilt-edged changes during the game, firing over in the 11th minute when clean through on goal, failing to convert Raphael Guerreiro's cutback late in the first half, and blasting over once again not long after the restart.
The visitors were handed a reprieve when Ljubomir Fejsa, arguably Benfica's weak link in defence after a number of nervous mistakes in the first half, handled in the box to gift Dortmund a penalty.
However, the Gabon striker's spot kick was right down the middle and at a comfortable height for goalkeeper Ederson, who parried the ball away without too much difficulty, and just minutes later, Thomas Tuchel had seen enough, replacing Aubameyang with Andre Schurrle.
Benfica were forced to repel a wave of Dortmund attacks in the latter stages but managed to hold on and claim a one-goal lead in the two-legged tie, which will be settled at the Signal Iduna Park on March 8.