Bayern made the best possible start to the tie, opening the scoring in the seventh minute when Toni Kroos fired a half-volley past the outstretched hand of Wojciech Szczesny.
© AP Photos
The Emirates Stadium was silenced by the early goal and Arsenal struggled to settle against their German opponents.
The Gunners failed to create any clear opportunities before the interval and could have conceded a third when Mario Mandzukic rose highest in the area but the striker couldn't find the target with his header.
The hosts improved considerably after Podolski's strike and Wenger reacted by introducing Tomas Rosicky and Olivier Giroud to the game, with the latter drilling a volley straight at Neuer just seconds later.
However, it was Bayern who snatched the crucial fourth goal when Phillip Lahm's low cross was scrambled into the net by Mandzukic.
Arsenal desperately searched for a way back into the game in the closing stages, with Jack Wilshere seeing a powerful effort blocked by Neuer but the visitors held on for a vital away victory.