The Gunners lead the top flight at its halfway point having accrued nine more points than at this stage last term, when they finished fourth.
And Walcott admits that with every win Arsene Wenger’s side are more certain that they will be top of the pile on May 11.
|9/2||Arsenal are 9/2 with BetVictor to win the Premier League|
"But we won't get ahead of ourselves, we know how crucial every game is. We've got to make sure we look after ourselves and hopefully teams will start to slip up.
"Just as long as we keep on believing, which everyone is and we've been doing from the start of the season.
"Hopefully everyone can stay fit, which is important to have a nice run to the end of the season."
Arsenal are undefeated over the festive period, having beaten West Ham on Boxing Day before Olivier Giroud’s strike overcame Newcastle on Sunday - victories Walcott classes as vital.
"I think you could see from the [final] 10 games last year that it was coming and that we had it in ourselves," the winger said.
"Everyone has been absolutely solid throughout the whole year and it was nice to see Laurent [Koscielny] back today as well.
"It's been two tough away matches in this period, I think a lot of people doubted that we were going to get results.
"It just shows that we've got great character in that dressing room and we're getting stronger and stronger every week. Everyone trusts each other, which is so important. Going into the New Year, we're very excited."