Arsene Wenger's side led the table for long stretches of the campaign, but injuries to the likes of Aaron Ramsey, Mesut Ozil, Theo Walcott and Jack Wilshere have seen the Gunners slip into fifth place behind Everton.
Arsenal is still on course to land its first piece of silverware for nine years after beating the Latics 4-2 in a shootout at Wembley on Saturday, and Arteta says the win will spur on the north London side to land a Champions League qualifying place.
"I'm very happy because we suffered a lot," he told Goal after the game. "I think that we deserve it, I hope that we can win. Now we have to focus on the Premier League, because we are fighting for a place in the Champions League next season. We'll have another chance to come back to Wembley next month.
"Many things happened over the last four weeks. We had eight injured players while our rivals were getting good results against us. That affected our confidence in an important moment of the season.
"We now have many weeks in which we have to turn up, and we hope to do it."
Santi Cazorla, meanwhile, believes the Gunners deserve to progress after a cup run that featured games against the likes of Tottenham, Liverpool and Everton - but must carry on fighting in the league to finish fourth.
“We deserve it due to all we did in FA Cup this season," he said. "The penalties, all the suffering of the match... I'm so happy I helped the team. That's my job, to help my teammates. Now we have to recover, and on Tuesday we have another important match [against West Ham]. Both are so important.
"To win a trophy after nine years... and the Champions League is very important too. We have many finals to win before the end of the season, and West Ham is the next one."