Following an intense build-up to the last 16 first leg, the north Londoners got off to worse possible start when Toni Kroos netted a seventh-minute opener.
Things soon worsened at the Emirates Stadium when Thomas Muller struck, before a reinvigorated Arsenal returned for the second half.
Lukas Podolski reduced the deficit against his former club, but Mario Mandzukic gave Bayern a dominant-looking 3-1 lead heading into the second leg in Germany - much to Arteta's frustration.
"We are very disappointed with the result," the Spanish midfielder said. "I think for them to be 3-1 up is not good enough from us. We made some mistakes that cost us a lot in the game. That was the difference.
"We have to go for it. We know it is very difficult at the moment but you never know in football. We may score early in the game and if we keep believing, as we did in the second half."
Arteta labelled Mandzukic's goal a "killer" for the Gunners, especially as it came shortly after substitute Olivier Giroud was denied by Manuel Neuer.
"I think we put everything into the game," the midfielder added. "I think we started the game well and the first time they got around the box they scored.
"After we conceded a sloppy goal at a corner again. Once it is 2-0 to a big team like this, it is a difficult game to play.
"We kept trying, kept believing and we got in the game through Lukas' goal and we kept trying to get the goal. We took some risks and we paid for it on the break."