Arsene Wengers's side topped the table earlier this year but have fallen away and now face a battle for fourth place to secure UEFA Champions League football next season for the 17th year in a row.
Defeat to Bayern Munich in their last-16 tie ended Arsenal's participation in this season's competition last month but Germany international Podolski is confident they can still secure their first silverware since 2005.
First they must defeat defending champions Wigan, who dumped Manchester City out in the last round, when they face them at Wembley on Saturday.
"When you play for Arsenal, every year you fight for the title and this year we were close to winning the league," he said.
"Nearly the whole season, we were top of the league and now the last matches were not so good. We lost a lot of matches and we lost a lot of points so now we are a lot behind.
"It is difficult to look for the Premier League title, but we have a big chance for the FA Cup and we must keep that chance. It is the biggest cup in the world and a big tradition.
"After a long time it is an important thing for the club and the players as well, so let's see what happens first on Saturday and then if we go to the final."