Christian Pulisic pounced on slack Bayern defending to delight Signal Iduna Park with the opening goal after 12 minutes but Robert Lewandowski ensured his former club's advantage was short-lived.
Pierre-Emerick Aubameyang produced a sublime finish on the end of Ousmane Dembele's pass 19 minutes from time before a late penalty box scramble resulted in an own-goal from Roman Burki.
Dortmund goalkeeper Burki saved from Joshua Kimmich in the shoot-out but opposite number Sven Ulreich denied Sebastian Rode and Marc Bartra to ensure the Bundesliga champions lifted the first silverware of the season.
"I'm not happy with the result. Two minutes before the end we have to win and we have not done that," said former Ajax boss Bosz after his first competitive game in charge.
"We were in the lead three times, even in the penalty shoot-out. In the end we have not won and I'm disappointed.
"As a coach I'm not satisfied with a defeat. On the first goal, our high press worked out and we did really well. Afterwards Bayern were better, with more chances.
"We were happy with 1-1 at the break and we played better in the second half because we were more compact and lost the ball fewer times – before it was too many ball losses and then there was too much space for our pressing [to work] because we were not compact.
"If we do that better, it will be difficult for Bayern Munich; if we lose the ball so often, we put more pressure on ourselves."
Mario Gotze was absent from Dortmund's matchday squad, having returned to action during pre-season on the back of five months due to a metabolic disorder.
Bosz confirmed the Germany forward had suffered a minor setback and that patience was the key as he sought to get his career back on track.
"As I said, we have to be careful with him," the 53-year-old added. "He's had a five-month break.
"After two steps forward he has taken half a step back and he has to move two-and-a-half steps forward."