The Gunners were up against it when Toni Kroos opened the scoring in the seventh minute and they were two down in the 21st when Thomas Muller pounced on poor defending to stab home.
Arsenal were booed off at half-time and responded at the start of the second half with a header from Lukas Podolski narrowing the deficit against his former club.
But after substitute Olivier Giroud fired a great chance straight at Bayern keeper Manuel Neuer from point-blank range, the Germans cut through the home side again on 77 minutes as Mario Mandzukic restored their two-goal advantage.
It left the home fans with a bitter taste in their mouths and Wilshere acknowledged the north Londoners were made to pay for a slow opening.
"We didn't start great and against good teams you get punished," the England midfielder said.
"In the second half we stepped it up and showed we could play but it's unfortunate we didn't do that in the first half.
"We could have got one with Giroud but it was a good save and they nicked another one. It'll be tough away but we're not out of it yet."
Gunners boss Arsene Wenger had hit out at critics of him following their FA Cup exit at home to Blackburn, but there was to be no redemption as the Germans were superior in every department.
Meanwhile, Joao Moutinho grabbed the only goal and put on a midfield masterclass as twice former winners Porto grabbed a 1-0 home win over Malaga.