The Blaugrana, who travel to Mestalla on Saturday, have often looked to Los Che to strengthen their squad in recent seasons.
Franck Ribery, Thiago Alcantara and Arjen Robben were all on the scoresheet as the Bavarians moved back to the top of the Bundesliga table with victory on Wednesday.
The BVB star ensured Der Klassiker bragging rights went to Thomas Tuchel's men this time round, as Die Roten were unable to respond to the early setback.
Sergio Ramos and Karim Benzema's goals in the first half left the Concacaf club - who had a penalty saved - trailing before Gareth Bale and Isco cleaned up in the latter 45 minutes.
The Athens outfit bagged the full three points, but will have to settle for a spot in the Europa League round of 32 nonetheless.
The Republic of Ireland international struck in extra time to secure his side their fifth Major League Soccer championship.