The Gunners beat Bayern 2-0 in Germany on Wednesday but crashed out on away goals after levelling the aggregate score to 3-3.
Despite their exit, the Spaniard feels that his side deserve praise for becoming only the second team this year to beat the Bundesliga leaders at home.
"We gave absolutely everything we had in the game. We knew we had conceded three goals at home and it was going to be difficult, but we kept going and we kept believing," Arteta told Sky Sports News.
"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. We scored a second one at the end and maybe with 10 minutes more we would have had a better chance.
"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."
Arsenal will now turn their attention to facing Swansea City at the Liberty Stadium on Sunday.