The north London club currently occupies top spot but lost its last league outing against reigning champion Manchester United and faces another stern test in the form of high-flying Southampton on Saturday.
Szczesny, though, says Arsenal's form over the last six months is proof the club can end its wait for silverware, whether manager Arsene Wenger brings in reinforcements in January or not.
"I don't want to say nobody can challenge us for a trophy but if we are at our best we will win a trophy and that is all that matters," the goalkeeper told reporters. "I wouldn't say we have proved everyone wrong and passed all the tests. I will say that when we get our hands on a trophy in May. I think what is important is what we believe in and we believe that we are good enough with the players we have got and the work we are doing to win the Premier League.
"We have shown that to people over the last six months or so. We know that if we just carry on doing the same thing we will show people that this team and these players are good enough to win a trophy without any extra additions or extra work that people are talking about. We certainly believe we can do it. We need to make sure we continue doing what we have been doing recently and keep getting the results."
The Poland international has also backed Arsenal to beat any other team in the world, and says the squad is full of confidence following on from a storming end to the 2012-13 season, which began with a win over Bayern Munich.
"It is quite an enjoyable time to be around the training ground,” he added. "When results go your way everything seems to be in the right place and everyone in a great mood so we are all happy. But what we want to do is make sure we keep it going and not just rest on our laurels and think we have done it now."
"There is no reason not to [have that belief]. When you look at the last few months almost 12 months really because it is not only this season we’ve been doing well. It was the last two or three months of last season, where we went unbeaten for a few games. We are confident that we see the quality is there and we know that when we play at our very best and do it consistently well, we can beat anyone basically."
The goalkeeper says his team is at its best right now, with everybody in the squad keeping a consistent level of play.
"A couple of years ago there was a case of not being consistent enough individually as a team but now you can see that whoever comes into the team, whoever comes in does a fantastic job. We are at our best at the right and it seems to be bringing results."