Goals from Olivier Giroud and Laurent Koscielny secured a 2-0 victory but Arsene Wenger's team were eliminated on away goals after a 3-1 defeat in the first leg at the Emirates Stadium.
Arteta praised his team-mates for a "terrific" result but conceded that it also served to highlight their frustrating failure to win easier games.
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"The way we conceded the goals against Spurs - we were dominating the game but we ended up losing it and that's frustrating.
"We put some things right last night. To get a clean sheet here is very difficult. You look at the side they have got and they are probably the best team in Europe.
"So to come here and get the result we did last night was terrific. We showed that it really matters for us and we showed great spirit.
"I am convinced we can compete but those details against big teams are massively important and for me the frustrating thing is when we lose games and the opposition does very little to win.
"If they play better than you as Bayern did in the first leg for 25-30 minutes, you put your hands up and say, 'Listen, terrific team'.
"But some of the games we have lost this season, that wasn't the case and that's why we have to cut it out straight away because it is going to be difficult to get what we want."
Arteta denied that the range of different nationalities in the Arsenal dressing room made it difficult to breed a strong team spirit.
"We do really care," he insisted. "We have a lot of different nationalities here and sometimes that is a difficult thing to manage. In the changing room, I haven't seen that problem.
"They are all at it, the boss is at it and we are going to keep fighting until the end. We get criticised and some days we really deserve it and need to be hit on the back. For last night, we deserve better.
"The criticisms come sometimes and the line is very thin when you win and lose. The perspectives of things are very different but we have got the characters and the desire.
"I would like to say how many teams would do what we did last night when we lost 3-1 at home to Bayern Munich? Let's be consistent - that is the main thing for us now.
"For us, Champions League football is massive. But not only for us, for the club and the fans. This club cannot afford to be out of the Champions League. That is the target and we cannot afford to miss it."