After suffering a four-game winless streak, the Blues triumphed over their London rival on Saturday, and the defender is confident of further progress under Jose Mourinho.
Mourinho's side had been under pressure going into the game following consecutive defeats to Everton and Basel. Chelsea, though, responded in confident fashion, briefly returning to the top of the league thanks to goals from Oscar and John Obi Mikel, whose strike marked the first time that he had found the net for the Stamford Bridge club.
Cahill was delighted to return to winning ways and was quick to dismiss talk of a crisis.
"Two dodgy results back-to-back got everyone talking," Cahill told the club's official website. "When you are a big team and you lose two games it gets talked about as a crisis. It was not ideal, but we didn't feel that way.
"We looked a bit tentative first half, but the second half we created a lot more chances, put Fulham on the back foot and we looked a better side. It was much more like us and that is why we got our joy.
"The second half was how we should be playing at home. It takes time when new players come in but we have a great squad now, a proven manager and I am sure we will do well.
"At the start of the season I thought we were playing some great stuff, but it does take time, because we have players needing to adapt to the league and to adapt to being here."
Chelsea bolstered its squad significantly in the transfer window, and England international Cahill believes that the added strength in depth can only be a positive thing.
"There is competition for places with world-class players in all positions, and everyone is aware of the squad we have," Cahill said. "So we know that if we play poorly we will surely be out the team.
"That is healthy, and we do play many games so having a good squad is important. No one is complacent, and that is a good thing."