The first goal made for a nervous evening for the home crowd, although Bayern gradually took control of the game and didn't allow any more dangerous chances for the rest of the half.
Laurent Koscielny (R) added the second in the 86th as Santi Cazorla sent in a corner with Koscielny beating Javi Martinez to the header and sending the ball low inside the right post.
Bayern Munich barely staved off an improbable comeback on Wednesday, eliminating Arsenal from the Champions League despite a 2-0 loss at home.
The hosts then dominated much of the second half but were limited mostly to long-range attempts. Arsenal then added more suspense into the match by netting the second as tension flared up from both sides.
Bayern are four-time winners of the competition and were runner-up twice in the last three seasons.
Arsenals exit from Champions League means England are without a team in the quarter-finals for the first time in 17 years.
Olivier Giroud gave Arsenal an early lead in the third minute. Walcott raced into the penalty area and sent a hard, low ball through the legs of Dante and across the goal for Giroud to drive it into the roof of the net.