The humiliating defeat sees Arsenal exit the competitions at the last-16 stage for the seven consecutive season, with the German giants running riot in the second half after Laurent Koscielny's red card.
Theo Walcott gave Arsenal an early lead as the hosts started at a frantic pace and Bellerin hopes similar performances will make amends both the end of the season after droves of fans left early.
"In the end of the day we play for them and we want them to be happy," Bellerin said of the supporters.
"We know we have the capacity and potential to do well, we've got the players and everything, we just need to think about every game the way we started today.
"If we play that kind of football then we can be up there.
"[We] say sorry to the fans because we need to do better than that."