The Barclays Premier League leaders had looked good value for a share of the spoils from Tuesday night's Group F encounter after an equaliser by Giroud just before half time helped them recover from a slow start when Henrikh Mkhitaryan put the well-drilled Germans ahead.
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 instead 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 to cap a clinical counter-attack in the 82nd minute.
With Napoli beating Marseille, three teams are now locked on six points. Arsenal head to Dortmund in a fortnight for a match which will have a significant weighting on their qualification hopes for the knockout stages.
Nevertheless, Giroud insists one below-par performance will not derail what has so far been a campaign full of promise, with Arsenal out to consolidate their position at the top of the table when they travel to managerless Crystal Palace on Saturday lunchtime.
"We lost (against Dortmund), but we have won almost every game - just (Aston) Villa and (a draw with) West Brom, so it is not the time to be really worried, but we want to come back Saturday and win the game," the France striker said.
Giroud has proved pivotal to Arsenal's attack this season, scoring seven goals as he leads the line in front of a formidable midfield now bolstered by a fit-again Cazorla who could line up alongside Mesut Ozil once again at Selhurst Park.
However, the Gunners forwards insists there is no concern of burnout.
"I feel good. Maybe I will have some rest next week, for the Capital One Cup, I will speak to the boss," he said.
"Over the years I have got used to playing a lot of games and I am okay with my body. I don't want to find a excuse."
Giroud believes Arsenal must learn to retain focus for the whole match, instead of being caught cold when they should have been concentrating on closing out a positive result.
"We finished the game better than them and that is why I am really disappointed," he added, "because if we couldn't win the game, we didn't have to lose it - maybe draw. In the Champions League, it is like that, about the details."