Arsenal earned a big scalp at Stamford Bridge over the weekend when they defeated Chelsea 1-0 to reclaim top spot from Manchester City.
Mikel Arteta's side will spend the next six weeks at the top of the table if they beat Wolverhampton Wanderers this weekend before the league goes on a break for the FIFA World Cup.
Arsenal have not won the Premier League since 2004 and have not played Champions League football since the 2016/2017 season.
After missing out on a top four spot in the final weeks of last season, Arsenal's main aim was to go one step further this term but Zinchenko believes they can make the most of their bright start and achieve even greater success.
"It starts in the dressing room and it's on the pitch as well," Zinchenko, a four-time Premier League winner with Manchester City, said following the win at Stamford Bridge.
"We are trying to push each other, we are trying to help each other. I think this is the right way to do it.
"We have a dream, we believe that we can achieve something this season. Let's see what is going to happen.
"I can feel that this group of people, they are special. Starting from our staff and then all the people on our training ground, our fans, we believe in ourselves, every single game.
"I got used to hearing that Arsenal were top four. I would say that I really want to kill and destroy this stereotype.
"We need to look further, and I think this team, this group of people, the fans, they deserve more. That's what we need to do. We need hard work in this and let's see at the end of season.
"You ask me if we can challenge Manchester City. Of course, they are an amazing team, but you never know what can happen in football. We need to go step by step with every single game."
