The 23-year-old scored a hat-trick as the Gunners came from 4-0 down to defeat the Royals 7-5 at the Madejski Stadium on Tuesday and progress to the quarter-finals.
"We showed everyone what we can do," Walcott is quoted as saying by the BBC. "We never gave up and we are absolutely knackered.
"We had to be very direct in our play, and Jason Roberts caused us problems all night, but we just showed our character, and a lot of young players gained good experience tonight.
"Hopefully we can take that into the Premier League. We were good going forward, but maybe we were not great defensively at times."
The draw for the quarter-finals of the League Cup will take place on Wednesday.