Beleagured boss Gerard Houllier, well aware of the importance of the clash, held a meeting with his players ahead of the weekend game, and Carragher is quick to praise the importance of he Frenchman's words.
Carragher said: "The message was really we had to work a lot harder, especially when we haven't got the ball, and get back to the attitude and commitment we showed when we won the three cups in 2001," he said. "I think the lads responded to what was said.
"In many ways, I think the performance proved what I said a few weeks ago. We seem to be able to compete against the big sides but struggle against others.
"You wouldn't believe we could go two home games against Charlton and Fulham without a win and then get a win at Old Trafford, but it seems to have been like that here for as long as I can remember.
"When I first broke into the side under Roy Evans I remember losing at home to Barnsley and then winning at Arsenal a week later. Until we put that right we're not going to challenge for the title.
"I think the game against Middlesbrough next week is going to be totally different. I know a lot of people will be looking at what happened and asking why we can't get results like that at home.
"The onus is on us to break sides down far more than it was on Saturday and that's when we've struggled sometimes.
"We've got to build on what happened on Saturday. Although tactically it will be different next weekend, there's still a lot we should take from the win.
"The character, togetherness and spirit in the side was a lot better. It's frustrating we're not in a better position when we've shown we can win at tough venues such as Goodison, Stamford Bridge and Old Trafford. Now we've got to play a lot better in our last two home games."