Goalkeeper Sven Ulreich, deputising for the injured Manuel Neuer, was the hero for Bayern with penalty saves from Sebastian Rode and Marc Bartra after a 2-2 draw in regulation time at Signal Iduna Park.
Carlo Ancelotti's side twice found themselves behind in the curtain-raising Klassiker, with Lewandowski cancelling out Christian Pulisic's opener and Roman Burki's unfortunate late own goal meaning Pierre-Emerick Aubameyang was unable to call himself the match-winner.
Bayern came into the game on the back of four defeats in their past five friendly outings, including heavy losses to AC Milan and Liverpool, but despite a mixed performance against BVB, Lewandowski is confident they will eventually recover peak form.
"I'm very glad about the victory, it is the first title," he told Bayern's official website.
"We turned in a very good display. We made a few mistakes but it is the first match of the season.
"We have to step it up even more now. I'm sure everything will be going perfectly in a few months. We will surely be as strong as in the last or the second-to-last year.
"It was a different kind of preparation for us, but now we had a little more time to train, and I'm very happy that we won the first title."
Lewandowski's team-mate Thomas Muller added: "It was a good thing to win again, especially here in Dortmund – it is about prestige.
"It was great that Ulle gave us two saves in the penalty shoot-out, but there is always luck involved."