With a trip to Manchester United on the horizon, the 23-year-old wants the Gunners to show the same belief in the league, while Carl Jenkinson also hails the team's mentality
The Gunners found themselves 4-0 down after just 47 minutes but staged a magnificent fightback as Carl Jenkinson levelled the scores deep into injury time.
With the tie then sent into extra time at the Madejski Stadium, a stunning finale saw Arsene Wenger's men secure progression with another four goals being struck, including two from Marouane Chamakh.
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"We showed everyone what we can do. We never gave up and we're absolutely knackered at the moment.
"We didn't want the [first] half to end [after Walcott's goal] to be honest. We had a good 10-minute spell. We just had to go at them, that's all we had to do.
"We showed our character. There's a lot of young players in that dressing room and it's a great experience for them.
"Hopefully we can take that into the Premier League now.
"We scored seven goals away from home. We looked great going forward, perhaps we weren't great defensively at times, but the most important thing was getting the win and staying in the competition and being in the hat for the next round."
Jenkinson was awarded the Gunners' equaliser in normal time, although Walcott's earlier strike appeared to have crossed the line in the build-up. The defender was ecstatic to have helped his side secure a famous win with his first goal in an Arsenal shirt.
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The 20-year-old mirrors his team-mate's view that the Gunners' performance was uncharacteristically poor in the first half but insists that the nature of the group of players is to never give up.
"We just didn't start," the defender admitted. "It was a poor start from us. Like Theo said, it wasn't Arsenal and it wasn't good enough from all of us.
"No-one ever gives up when you're playing for Arsenal and we showed what we're all about. To come back from four goals down shows a slot of strength and a good mentality. We're delighted."