The Gunners earned an impressive 2-0 victory in Germany on Wednesday night, but the 3-3 aggregate scoreline was enough to see Bayern progress to the quarter-finals.
"It was close," Giroud told BeInSport. "We are disappointed for the fans who were behind us. We almost did the perfect game.
"We scored near the beginning and, after that, it was almost even but we managed to get the second. But then it was hard because the tank was empty.
"Bayern is a great team but we stood up against them. It's a shame because there were other goal situations but I think we made our fans proud."
Arsenal resume their Premier League duties on Saturday when they take on Swansea City at the Liberty Stadium.