The Gunners returned to the top of the table, one point ahead of Manchester City, after a late double salvo secured a 2-0 victory over Cardiff at the Emirates Stadium on New Year's Day.
Walcott - whose chip over the goalkeeper in stoppage-time made sure of the result after Nicklas Bendtner had crashed the ball home from close range with just two minutes left - maintains such determination shows Arsene Wenger's men have the character to last the distance this season.
"If you want to win titles then you have got to go to the end and the thing about this team at the moment is that we never give up," said Walcott.
"Every game that we win the belief gets stronger and stronger.
"A lot of people are not giving us enough credit at the moment, but we do not mind that.
"We will just carry on with our job, keep picking up three points, and see where we are at the end of the season."
Arsenal's victory, however, came at some cost with substitute Bendtner having suffered an ankle injury in the process of scoring, when Cardiff goalkeeper David Marshall caught him trying to make a save at point-blank range.
Lead frontman Olivier Giroud is also troubled by his own ankle problem, but could return to face Tottenham in the FA Cup on Saturday.
Bendtner is set for a scan on Friday, with initial fears he would be sidelined for several weeks.
Wenger, though, insists he will not be rushed into any hasty transfers, with fit-again German forward Lukas Podolski another attacking option, although he looked rusty when deployed in a central role against Cardiff.
Arsenal have been linked with a cut-price move for Fulham striker Dimitar Berbatov.
Walcott would always welcome some additional talent to boost what he believes is now a very "special" squad.
"To be fair that is for the manager to decide if he wants to go out and buy anyone, whoever will come to the club in January, if not it is going to make the club stronger anyway," the England forward said.
"It is always nice to have competition with players coming in, but January is such a difficult period in the transfer market to get players in, especially the top players.
"The boss has already said that he is looking for special players and we have got a lot of special players in that dressing room."
Bendtner, meanwhile, posted an update on Twitter thanking Arsenal fans for their messages of support.
The Denmark striker, now back in the first-team picture having seemed set to leave the club, said: "Have a scan tomorrow that will tell us more. So happy with the 3 points. Thanks for all your tweets and concerns. #fans #arsenalfamily #23"
Wenger hopes Bendtner will recover to continue to show the same commitment of recent weeks.
"Unfortunately he got injured, so I am really sad for that because we lose a striker of quality," Wenger said on Arsenal Player.
"He has quality, who questions his quality? Nobody.
"What you want from him is to be focused on giving absolutely 100 per cent and he is a great centre forward."