The Poland international has already agreed to join the Bundesliga champions on a free transfer this summer, much to the disappointment of BVB fans, who saw Mario Gotze leave for the Allianz Arena last year.
However, Lewandowski was afforded a standing ovation by the Signal Iduna Park faithful in his final home game for Dortmund on Saturday, against Hoffenheim, and he is now keen to repay their continued support by helping Jurgen Klopp's men claim the DFB-Pokal trophy in Berlin on May 17.
"I've learned a lot here in Dortmund and I know that I'm a better player now than I was four years ago," the former Lech Poznan forward told the Bundesliga's official website.
"But there's still a lot of football to be played. It was my last game in Dortmund but we still have two more matches to play in Berlin [against Hertha on the final weekend of the league season, and then Bayern in the German Cup final].
"These have been four fantastic years for me at BVB and they're not over yet. I've scored a lot of goals and experienced so many great moments here, both in the Champions League and in the Bundesliga.
"We won the league twice and hopefully we can win the DFB-Pokal twice as well. My story here isn't over yet."
Lewandowski could also finish as the Bundesliga's top scorer, as he currently sits alongside future Bayern team-mate Mario Mandzukic at the top of the charts, on 18 goals apiece. However, the 25-year-old hitman is reluctant to make it a priority.
"I don't want to put myself under any pressure," he said. "If I am top scorer at the end of the season, that would obviously be great . But it's harder to do that if you put extra pressure on yourself."