Carragher told liverpoolfc.tv: "We played very well. We got the early goals but I think the first 45 (minutes) was not the real Fulham.
"In the second half when they got the ball, I thought we started to get a little bit nervous ourselves but once Maxi scored (the fourth goal) I think the game was finished then.
"We've been playing well of late, confidence is good, we're looking forward to games and we wanted to start well. It doesn't always happen like that - I'm sure Fulham wanted to start well - it was just one of those days for us.
"When you win everything is enjoyable, and we're going in every day now with a smile on our face and looking forward to games. Confidence is high and we want to keep this run going."