The Blues are chasing a first Premier League title since 2009/10, but won the Champions League in 2012 and the Europa League in 2013.
Jose Mourinho has made it clear Chelsea are in a transitional period, but Cech is one of four survivors from his title-winning sides from his first spell in a squad full of talent and potential.
In recent years Chelsea have been out of title contention by Christmas, but Mourinho's men enter the Boxing Day bout with Swansea two points behind first-placed Liverpool, who lead Arsenal on goal difference and visit Stamford Bridge on Sunday.
"We are right up there - we are two points (behind)," Cech said.
"We don't depend on anyone else. If we win our games we have a big chance of winning the title so it's very positive and we know that we haven't hit the best level of this team yet. We are improving and hopefully that will show.
"We have won things in the past years which give us experience as a team so let's see if we can use that as our advantage.
"We have been in these situations many times fighting for the titles and playing under pressure before and after Christmas.
"Towards the end of the season you have a lot of pressure because you need to get results and hopefully our experience will pay off."
Chelsea have made a stuttering start and lost away to Everton, Newcastle and Stoke, while strikers Fernando Torres, Samuel Eto'o and Demba Ba - with five Premier League goals between them - are still to fully fire.
Too often, according to Cech, Chelsea have let their opponents off the hook, making their own lives difficult.
"We need to rediscover our killer instinct in the way that when we have chances to kill games off we need to do it," Cech added.
"You can't keep your opponent alive and it happened to us many times this year."
Chelsea drew at Arsenal on Monday after adopting a defensive approach which prompted home fans to label the visitors 'boring'.
Cech hit back at the criticism and could not resist pointing to Arsenal's eight-year trophy drought.
"Well in the end you count the titles," he said.
"If you have figure skating where you are judged by how nicely your routine is then obviously that is one point, but football is about points and goals and you need to get points to win trophies."