Gibbs returned from a six-week injury absence to aid Arsenal in becoming the first English side since Norwich in 1993 to win at Bayern.
However Arsenal were unable to avoid a third successive last-16 exit with a 2-0 victory, losing on away goals.
The sole focus for the season now is making up a five-point deficit on the top four - starting at Swansea on Saturday.
Gibbs told Arsenal's website: "We are right to get criticism but we're not right to be told that we don't have character in the team.
"We showed that tonight so anyone who has their doubts can have them, but the belief is there in the team. We just have to turn that into results and correct our mistakes and things will look positive for the team."
Gibbs, like many other players after the final whistle, had mixed emotions about the result of the game.
"You can only feel as good as the result but we're not through," he said. "I think the second goal might have come a bit too late. The boys gave everything as you can't fault the effort. It's disappointing."
Arsenal begin the scrap to make the top-four when they travel to Swansea on Saturday, the first of 10 crucial remaining Premier League games.