The Gunners sealed a 5-0 aggregate win over Fenerbahce on Tuesday night, three days after a convincing 3-1 win over Fulham in the Premier League.
And Walcott has now set his sights higher and called on his team-mates to challenge for the title and not settle for fourth spot.
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"We don't want to be [just] challenging for that fourth spot anymore. We want to challenge for the Premier League and that's our main goal.
"We know Arsenal's a massive club and where we need to be. What we need to do is get to the next level and compete a little bit more -- not just getting into the Champions League. The next step is to compete."
Aaron Ramsey scored in the first leg against Fenerbahce before hitting a double on Tuesday, and Walcott believes the Welshman could be key to Arsenal success this season.
"He ended the season last year very, very well, but this season he's just taken off and hit a new level. It's great to see because he deserves it after a lot of injury problems. I'm very happy for him."