In a frantic encounter between the Bundesliga champions and DFB-Pokal winners at Signal Iduna Park, it was Bayern who defended their crown in the third straight Supercup meeting between the Klassiker rivals.
Lewandowski broke Gerd Muller's long-standing Bundesliga record of 40 goals in a single season with the final kick of 2020-21, and he fittingly opened the scoring two days after the Bayern great's passing.
Muller put Bayern in complete control and, though Marco Reus' stunner gave Dortmund hope, Lewandowski capitalised on Manuel Akanji's error to seal victory.
Dortmund were their own worst enemy from the off, with Serge Gnabry, Muller and Kingsley Coman all squandering gilt-edged chances after sloppy play from Marco Rose's team.
Yet it was Neuer who made the first save of the match. Hailed as "the best goalkeeper in the world" by Nagelsmann, the Bayern stalwart stuck out a leg to spectacularly deny Reus.
Dortmund were caught cold from a free-kick as the intensity continued, Lewandowski's effort blocked on the line by Axel Witsel.
Youssoufa Moukoko, the youngest goalscorer in Bundesliga history, thought he had nosed Dortmund ahead in the 36th minute only for the offside flag to halt his celebrations, before Neuer kept out Erling Haaland.
Bayern made their fortune count, Lewandowksi floating into the area to direct a thumping header beyond Gregor Kobel.
Lewandowski was involved again as, four minutes after the break, Bayern doubled their lead, the striker doing enough to prevent Kobel from collecting Alphonso Davies' cross, with Muller turning home.
Dortmund seemed to have an immediate lifeline when Haaland raced through and slotted home, but the offside flag again came to Bayern's rescue.
Bayern's lead was cut in sensational style after the hour, Reus curling a sumptuous first-time strike into the top-right corner.
Despite the backing of much of the limited-capacity crowd, Dortmund's comeback hopes fell flat 16 minutes from time as Corentin Tolisso pressured Akanji and Lewandowski picked up the scraps to round matters off.