The England forward was delighted to have "shut people up" by clinching qualification for the Champions League at the expense of Tottenham, but wants his side to go one better
The Gunners beat Newcastle 1-0 thanks to a goal from Laurent Koscielny, and with that victory enough to see off the challenge of rivals Tottenham, it means they have secured a place in Europe's top club competition for a 16th consecutive season.
But Walcott, who believes beating Bayern Munich earlier in the season was a turning point, wants his team-mates to push on and challenge at the very top of the division in 2013-14.
"If we can beat Bayern we can beat anyone, that's what we went in to every game believing," he told Sky Sports.
"We just need to do it on a consistent basis from the start and hopefully instead of these celebrations for [making] the Champions League we can challenge for the title, that's our next step, our next goal.
"A lot of people didn't think we were going to make the Champions League today, I'm glad we did fantastically today and shut a lot of people up.
"I think we had to prove people wrong, we had to get something out of each other that we hadn't really done, we had a disappointing result against Tottenham and it's always not nice when you lose against them, but we bounced back well.
"I've got to say I'm very proud of everyone in that dressing room. We worked hard, we've had some ups and downs, everybody showed tremendous belief and that's what we're made of here at Arsenal."
Koscielny's goal won the match for Arsene Wenger's side, and he was pleased to have made such a vital contribution.
"It's important, my goal, but the team played very well, it was good because we are good players and want to play in the Champions League, it's a good competition and we did our job," he said.
"It's very important for the club to play CL, so we did the job well."