Toni Kroos and Thomas Mueller both struck inside the opening 21 minutes to put the visitors in charge, before Lukas Podolski halved the deficit just after the break.
However, Arsenal were unable to complete a stirring comeback despite substitute Olivier Giroud’s best efforts, and Mario Mandzukic sealed a 3-1 win in the 77th minute.
Walcott, who was kept quiet by a menacing Bayern defence, says the Gunners now have nothing to lose and can play with freedom in the return leg at the Allianz Arena on 13 March.
“We always felt that if we got the goal they might stumble a little bit, but they dealt with it quite well and they got that killer blow,” said Walcott.
“We can’t be down in the dumps about it, we’ve got to try and take the positives and learn from it. I thought we fought quite well and battled for each other.”
He added: “We need not to dwell on this result and when we go over there we’ve got to be positive.
“Last season against AC Milan we were 4-0 down [at this stage] and we did very well at home [winning 3-0].
“We’ve got to show what we can do and play without any fear because we’ve got nothing to lose.”
Lukas Podolski, who spent four seasons at Bayern, was not surprised by the Bundesliga leaders’ impressive performance.
“We must score three goals in Munich and this will be very difficult,” said the German striker.
“I think we came out well for the second half; when we scored it was nearly 2-2 and would have been another match, but it finished 3-1. We put on good pressure in the second half but we must switch on from the beginning.
“It was no surprise because they have played a very good season in Germany and they are a good side.
“It’ll be very difficult but we have a chance when we go to Munich and we will fight.”
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