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A clinical strike from Kroos opened the scoring after seven minutes and when Mueller made it 2-0 midway through the first half the visiting fans were already in party mood.
Lukas Podolski scored against his old club as Arsenal battled back after the break but Mandzukic bundled in a third Bayern goal in the 77th to seal their first ever win in London.
The Bundesliga leaders, trying to make amends for their heartbreaking penalty shootout loss to Chelsea in last year's final, toyed with Arsenal for 45 minutes and, though given a real scrap after that, are overwhelming favourites to reach the last eight.