The Gunners had looked good value for a share of the spoils from the Group F encounter after recovering from a slow start - Henrikh Mkhitaryan put the well-drilled Germans ahead - to level before the break through Olivier Giroud.
Indeed, if a sublime 20-yard curling effort from substitute Santi Cazorla had flown just a couple of inches lower instead of crashing off the angle, Arsenal may well have recorded what would have been a hard-earned victory over last season's beaten finalists.
However, Arsene Wenger's side were left with the disappointing taste of defeat for only the second time this season when Robert Lewandowski - somewhat fortunate to still be on the pitch after earlier catching Laurent Koscielny with a stray elbow - converted following a clinical counter-attack in the 82nd minute.
Napoli's victory over whipping-boys Marseille leaves three teams now locked on six points and Arsenal now face a tough trip to Dortmund in a fortnight, where another loss would put their qualification hopes for the knockout stages in serious jeopardy.
Wilshere, though, insists Wenger's men have the strength of character to pick themselves up and go again when the Barclays Premier League leaders head to bottom club Crystal Palace for Saturday's lunchtime kick-off.
"It is never nice to lose at home, but it is not the end of the world," said Wilshere, who was substituted in the second half as a precaution after aggravating his ankle.
"We can't let this result affect us now, especially in the Premier League where we are on a great run,
"It will be tough away at Palace, it always is, they will fight for every ball and it will be a different game from Dortmund, but we still have to bounce back straight away."