The Gunners headed to the Allianz Arena on Wednesday hoping to pull off a European miracle, with a 3-1 defeat in the first leg of their last-16 encounter leaving them with a mountain to climb.
Arsene Wenger's men came agonisingly close to completing a remarkable fightback, as they ran out 2-0 winners on the night.
The away goals rule means they will be missing from Friday's last-eight draw, but Arteta feels Arsenal offered the perfect response to those who continue to write them off.
He told Sky Sports: "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.
"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."
Carl Jenkinson also believes that Arsenal should be looking to use their performance on German spoil to spark an end-of-season surge - with the Gunners still scrapping for a top-four finish in the Premier League.
He said: "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."