The Bundesliga champions outclassed Milan on Tuesday but were made to work harder for their win against an under-strength Real, who started without Gareth Bale or James Rodriguez.
Bale and Rodriguez scored for Rafael Benitez's side in the victory over Tottenham a day earlier and, in the absence of that duo and the injured Cristiano Ronaldo and Karim Benzema, the likes of Jese Rodriguez, Lucas Vazquez and Denis Cheryshev struggled to muster much attacking threat.
Thomas Muller's deflected effort was Bayern's best chance of the first half while substitute Lewandowski missed a sitter early in the second.
But just when the final was set to go to penalties, Lewandowski showed his poaching instincts to turn home Douglas Costa's wonderful cross and see Bayern into Sunday's DFB-Pokal meeting with Nottingen off the back of a win. Real complete their pre-season at Valerenga on Sunday
