The reigning European champions were under the cosh early in the match as the Gunners started brightly but a red card to goalkeeper Wojciech Szczesny late in the first half changed the direction of the contest.
Arsenal had their chances during the first half and would have been left to rue that profligacy when Szczesny was given his marching orders for colliding with Arjen Robben in an aerial contest when the Dutchman was the last man.
The decision from referee Nicola Rizzoli appeared to be harsh as the ball was clearly moving away from the direction of the goal but luckily for the Gunners, left-back David Alaba side-footed his effort into the past of the post when sending substitute keeper Lukasz Fabianski the wrong way.
Early in the second, however, Bayern broke Arsenal's resistance when skipper Philipp Lahm found Kroos just outside the area and the 24-year old produced a sublime curling effort which planted itself nicely in Fabianki's top left corner.
Arsene Wenger's Arsenal were doing their best to hold out for as long as possible but finally Pep Guardiola's Bayern added a second when Lahm chipped into the area for substitute Thomas Muller to score with a diving header in the 88th minute.
It was the first time ever in a Champions League knockout stage match that both sides had missed penalties but in the end, Bayern got the job done, as they did last season when beating Arsenal 3-1 in London.
The Gunners will now be looking for a repeat performance of last campaign's second leg when they got the better of Bayern 2-0 in Munich if they want to progress to the quarter-finals for the first time since 2010.
2013/14 UEFA Champions League - Round of 16 first leg
Arsenal (ENG): 0 (red card: Wojciech Szczesny 37', missed penalty: Mesut Ozil 8') Bayern Munich (GER): 2 (Toni Kroos 54', Thomas Muller 88', missed penalty: David Alaba 40') Emirates Stadium, London, England - 59,911
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