Arsene Wenger's team had been run close by the Championship club but Walcott emerged from the bench to strike late and send them into the last 16.
The Gunners have now bagged eight goals in their last two games having beaten West Ham 5-1 prior to the fixture on the south coast, and Walcott has urged his team-mates to believe in their own ability.
"I am very confident in the team, this team just needs to realise how good we actually are and not read into too many bad thoughts," Walcott told reporters.
"We have a very good team, very young and great on the eye at times. We can be cold at times as well, but we are starting to learn and we are starting to get back to our best."
Despite Arsenal's testing campaign Walcott has scored 17 goals in all competitions, surpassing his best record of 13 in a season in 11 fewer matches.