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 told Arsenal Player: "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.
"But as long as we keep on believing, which everyone is there, and which we have been doing from the start of the season...
"Hopefully everyone can stay fit, which is important, to have a nice run towards the end of this season now."
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 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 goals. 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 free-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."
Arsenal's win on Tyneside 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 win at West Ham 3-1 on Boxing 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 that we have got great characters in that dressing room and we are getting stronger and stronger every week.
"Everyone trusts each other, which is so important in the dressing room now, so going into the new year, we are very excited now.
"You could see from the 10 games last year when we had that run that it was coming and we had it in ourselves.
"It was an absolute battle. Newcastle, it was always going to be tough here. We knew from the start it wasn't going to be the prettiest game.
"But when you want to win the title, you have got to come to places like here and try to get a result, and we managed to keep a clean sheet, which is important."
Source : PA
Source: PA