The Gunners played with 10 men for the last 20 minutes of Saturday's 3-2 win over Swansea as the visitors threatened to come from behind, having won at Arsenal in the past two seasons.
The win means Arsenal are level on points with Manchester City at the top of the Premier League and follows a last-gasp victory over Burnley before the international break.
Most of Wenger's sides throughout his 20 years at the helm have been accused of not being able to grind out those results but Walcott feels it is proof that the current squad has matured and is ready to challenge for honours.
"The Burnley game, they were very well organised and we found it very difficult," he said. "But we all stuck in, showed patience and in time we got the win which was really important.
"Again against Swansea, we had to be diligent, be patient at times... but we still got the win even if it would have been nice to take the ball home on a personal note. But that's not the case.
"The Burnley match was a game where we had to grind it out. This game, when we went down to 10 men, we tend to lose, to be honest. There's some things that are slowly getting better.
"I don't look at the table. I genuinely don't. I feel all we can do is control ourselves and not really hope that other teams get results on our behalf. We can only control ourselves, if we play well we win games and that's all we can do.
"We've been in this situation (top of the table) before. We started off this season before than we have recently. There's always some sort of blip but hopefully we're not going to have that this year.
"With the squad we have, if we keep players fit, with players coming back, then who knows... You can't really call the Premier League this year. It's so unpredictable."