Despite an impressive 2-0 victory in the Allianz Arena, Arsene Wenger's men bowed out of Europe after they were unable to recover from a 3-1 first-leg defeat in London last month.
The Gunners must now lift themselves for Saturday's trip to Swansea, where victory would keep alive hopes of chasing down Chelsea and perhaps even fierce rivals Tottenham for a place in the top four of the Premier League.
"It has to give us confidence, but we must keep our feet on the Earth as well because we haven't been on the consistency [level] that is required at this football club," said Arteta, who was captain on Wednesday night as Thomas Vermaelen was left on the bench.
"On Saturday, if we are not at it, we are going to face a really difficult team which is Swansea away from home."
Arteta added: "For us it (Champions League) is massive, for the club and the fans, we cannot afford to be out of the Champions League.
"You never know in football. We have to focus on us. At the moment it doesn't depend on us, but the only thing we can do is win every game."
Arsenal may have been facing a 'Mission Impossible' in Bavaria, but showed the desired mindset as they took an early lead through Olivier Giroud.
After weathering long spells of Bayern pressure, Laurent Koscielny headed in a second to level the tie on aggregate with four minutes left, but it was not enough as Arsenal went out on away goals despite becoming the first English team to beat Bayern at home in Europe since Norwich won at the Olympiastadion in the 1993 UEFA Cup.
Arteta is in no doubt Arsenal can cut it at the highest level, but admits they have too often proved the architects of their own downfall.
"I am convinced we can compete, but those details against big teams are massively important," said the Spaniard. "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 is why we have to cut it out straight away because it is going to be difficult to get what we want."
Arteta continued: "Some of the critics are right because when you concede three at home, you can expect something different.
"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 against Bayern.
"You look at the side they have got here, they are probably the best team in Europe, so to come here and get the result we did is terrific."
Arsenal have often been criticised for a supposed lack of leadership within the squad.
Arteta, though, disagrees.
He said: "We have got the characters and the desire, we do really care.
"We have a lot of different nationalities here and sometimes that is a difficult thing to manage.
"However, in this changing room, I haven't seen that problem and we are going to keep fighting until the end.
"We get criticised and some days we really deserve it. After this result, we deserve better."
Wenger has a few selection decisions for the trip to Wales, notably whether to allow Arteta to retain the armband.
Goalkeeper Wojciech Szczesny was left back in London, with Wenger revealing the Poland international had been "mentally affected" by the number of games he played this season.
Despite not playing a first-team match in more than a year, Lukasz Fabianski produced a solid display, making some smart saves.
Arteta said: "There was a lot of pressure on him, mentally as well, but he came in and was fantastic.
"Lukasz has been working really hard, he has done it here, so why can he not do it in the following games?"