The Gabonese attacker found the net seven minutes from time in the last-eight clash at Eintracht Frankfurt on Tuesday, ensuring the UEFA Champions League runners-up progressed a round further than last season.
Jurgen Klopp's side are also in the last 16 of the UEFA Champions League and sit third in the Bundesliga.
And Aubameyang believes his side have the quality to be competitive on all fronts.
"There’s a lot of quality in the squad," he told the club's official website. "We can achieve much more if we stick together.
"If we play with a knife between our teeth and fight until we drop we can go far in every competition."
With Dortmund 17 points behind leaders Bayern Munich in the race for the Bundesliga title, coach Jurgen Klopp knows the importance of success in cup competitions.
"We are very happy, it was an important win,” he added.
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