The Londoners plunged to their ninth league defeat of the season on Monday, losing 2-1 at leaders Leicester City to slip to fifth from bottom.
"We will all have to take our responsibility. If you are a big player, and paid like a big player, you must play like a big player and behave like a big player," Fabregas told a Facebook question and answer session on Tuesday.
"I am not saying you can't have a bad season and bad games. We all have big players and small players but the attitude must be spot on.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"We must always be at the top of our games, even when it's not, and the behaviour has to be better than what we are seeing right now from every single Chelsea player."
The Spain midfielder refused to blame the squad's problems on a lack of self belief.
"Confidence is important until a certain point," said Fabregas. "At the end of the day, as a player you have pride.
"We cannot win the Premier League right now but we must do better. Right now is not the time to think of where we might finish, we've been saying it almost all season.
"I don't think about anything apart from wanting to beat Sunderland (on Saturday), play my best and that's it," added Fabregas.
"Right now no one is easy (for us). We've put ourselves in this position and we must now prove that we can start winning."
Fabregas' former Arsenal team mate Thierry Henry also took part in the question and answer session and the Frenchman believes the blame for Chelsea's poor form should be directed more at the players than at manager Jose Mourinho.
"It seems like it's difficult for them and yes, you can have a go at the boss, but I also think you can have a massive go at the players," said Henry.
"It is time for the players to be professional about it, where is your pride?
"With the quality in that team...how can you not beat Bournemouth, Crystal Palace, and it's no disrespect to any of those teams, at your place?," added Henry.
Chelsea lost 2-1 at home to Palace in August and were beaten 1-0 by Bournemouth at Stamford Bridge earlier this month.
(Writing by Tony Jimenez; Editing by Toby Davis)
