Goals from Toni Kroos, Thomas Muller and Mario Mandzukic secured the victory for Bayern, a result which the Gunners must now overturn in the last-16 second-leg tie in Germany on 13 March.
But Arteta, while insisting Arsene Wenger's men will keep believing, is fully aware of the challenge that lies ahead.
"We are very disappointed with the result," said the Spanish midfielder.
"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 we must keep believing, as we did in the second half.
"Once it is 2-0 to a big team, it is a difficult game to play. We kept trying, kept believing and we got in the game through Lukas' goal and kept trying to get the goal. We took some risks and we paid for it on the break."
Former Bayern Munich player Podolski added: "The result speaks volumes. Especially in the first half, Bayern put us under huge pressure and played very well.
"We started well after the break and deservedly pulled one back, which is what we said we'd do at half time. But we've lost 3-1. It'll now be very difficult for us away from home."