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 told ITV1.
"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 on Monday hit out at critics of him and his side following their FA Cup exit at home to Blackburn, but there was to be no redemption as the Germans were superior in every department.
Wilshere refused to lay blame on the Frenchman, however, adding: "I don't think it's anything to do with the manager.
"He puts us on the pitch, it's down to us to perform and get the best out of us every week and a manager can put us on the pitch and motivate us but we have to do it when we're on the pitch and at the moment we're starting badly and getting punished for it.
"I think the players will take responsibility, we're man enough to take it. The boss has been here for 16 years and he's been doing a great job so you can't question him."
Former Bayern striker Podolski admitted that the Gunners have a tough task ahead of them now.
"The result speaks volumes. Bayern forced a lot of pressure in the first half, though we came out well in the second half and pulling one back was deserved. It will now be very difficult for us away from home." he added.
Winger Arjen Robben, who provided the cross for Mandzukic to scuff home Bayern's third, was thrilled with his side's strong start.
The former Chelsea wideman knew Arsenal would provide a stern challenge and believed making a positive opening would subdue a partisan Emirates Stadium crowd.
"We expected a very big start from Arsenal here at home with the fans behind them," 29-year-old said.
"I know from my time in England and we warned all of the players we have to be there from the first minute because there will be a storm going here at the Emirates Stadium.
"I think we did very well and of course if you score two goals you're in a great position."