The German side resoundingly triumphed 3-1 at the Emirates Stadium, a result which had many people writing the Gunners off before the second leg had even started.
Goals from Olivier Giroud and Koscielny gave Arsenal a 2-0 win at the Allianz Arena on Wednesday night but it was not enough to rescue the north London side's chances of winning a trophy this season.
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"Tonight, there is no place for regrets. We wanted to pressure them high and we were good. The second goal comes in late. It is as it is. We fought well, we tried to produce the play but there was a big team in front of us."
Giroud was equally disappointed, considering that Arsene Wenger's side came so close to taking an aggregate lead, but he is confident that the fans will be proud of their efforts.
"It was close," he noted. "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."
Speaking to Sky Sports, Mikel Arteta echoed Giroud's thoughts that an earlier second goal would have been perfect.
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"It was mentally a very difficult game to manage tonight. When we scored the first one they didn't look very convincing and we tried to take advantage of that.
"Coming here, not many teams win here. They have shown that they are a top side in Europe this season, so the lads deserve credit and the boss as well because he believed from the start that we could do it. We tried our best."
Like his team-mates, Carl Jenkinson highlighted the importance of their early goal and, although they were ultimately knocked out of the competition, he is adamant that the Gunners can be proud of their efforts.
The 21-year-old stated: "We got that early goal that we needed. We knew that it was important that we kept it tight at the back and we did that. We got the second goal and we really thought we could do it, but I am proud of the boys and the effort everyone put in.
"We were so close to doing it when, to be honest, everyone had written us before the game. We have come here, we have won 2-0 and come so close to doing it. I am very proud of the boys."