A hard-fought 1-0 victory at Newcastle on Sunday saw the Gunners leapfrog Manchester City into top spot once again with the campaign now at the halfway stage.
Manager Arsene Wenger is cautiously optimistic, but knows there is too far to go to make any bold predictions, and England international Walcott is remaining equally level-headed.
He said: “Every game that goes and we get wins, it’s more belief that we are going to win the league.
“But we won’t get ahead of ourselves. We know how crucial every game is and we have got to make sure we look after ourselves, really, and hopefully teams will start to slip up.”
There was little to choose between the sides at St James’ Park and a tight contest was eventually settled by Olivier Giroud’s 65th-minute header, a glancing headed effort from Walcott’s inviting free-kick.
It was the Frenchman’s 11th goal of the season, but his first since November 23, and it came after he had picked up a foot injury in a rugged challenge by Magpies defender Mike Williamson.
Walcott said: “When you play up front, you want to score. He hasn’t scored for a few games now, so that goal will give him a lot of confidence.
“You need your main striker to be scoring up front throughout. I didn’t think he touched it, to be fair, when I kicked it in, but I am happy for him because he deserved it.
“He works really hard for us up front all the time. He’s had a massive role to make us stay at the top of the league, so hopefully he can stay fit now and hopefully that’s not too much of a bad scar on his foot.”
The win was all the more pleasing for Wenger and his players because it was earned with a gritty performance which saw them having to defend for their lives amid a late onslaught.
The manager later paid tribute to a resilience his side has been accused of lacking in recent years, but one which has increasingly become a feature of their play having come from behind to beat West Ham on St Stephen’s Day.
Walcott said: “We have had two tough away matches in this festive period, so I think a lot of people probably doubted us that we were going to get results.
“But it just shows we have great characters in that dressing room and we are getting stronger and stronger every week. Everyone trusts each other, which is so important now, so going into the new year, we are very excited.”
Meanwhile midfielder Alex Oxlade Chamberlain believes Arsenal now have the “spine” needed to continue to push towards the title.
“You need to build a base and foundation,” he said. “It’s called the spine of the team and I think we’ve got that now, right from defence all the way up to the attack. We’ve got some really solid players and a lot of talented players all around them. Keeping players in and around the squad now for the last two, three years really is a foundation. Once you spend two or three years at a club, especially at Arsenal, because the boss is so consistent with his ways, you start to learn what’s needed and what’s expected.