Spurs finished above Arsenal for the first time in 22 years last term but the Gunners went on to pick up a third FA Cup in four years with victory over Chelsea at Wembley.
Many onlookers believe Mauricio Pochettino's side to be superior ahead of the first derby of the season on Saturday but Cech, who will start in goal at the Emirates, believes they must add some silverware to justify their recent resurgence.
"No I don't think so," he said when asked by Sky Sports whether there had been a power shift in north London over the last few seasons. "There is always one odd year where things can change, but the most important thing for us is competing against the other teams for the Premier League title and for the glory.
"We are not competing against Tottenham Hotspur. Two years ago when we finished above them we were disappointed because we didn't win the title. Our target is to win the title and be successful during the season, not necessarily looking at what Tottenham are doing."
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He added: "They've been there in the last few years so now I think they have to make sure they win something to show the progression. Success is ultimately winning trophies.
"We were under pressure the last couple of years, people talking about winning trophies, obviously we are disappointed not to have won the title but we won FA Cups. Although the season was not always what we wanted, we always had a trophy at the end of it."
With their contracts running out in 2018 it has been suggested that Arsenal should allow both Alexis Sanchez and Mesut Ozil to leave for the good of the team.
But Cech doesn't believe that to be the case.
"They are an important part of the team," he added. "With the quality of Alexis and Mesut you are always happy to have players like this in the team.
"But you need them to perform, you need to feel they are committed to the case, I have to say they come every day, they work and as a team we feel that - although there is a lot of speculation, their contracts are different to everyone else's - you can see they are committed to the team.
"They train well and we hope their form will help us reach the target."