Reds struggled for much of the game, but came out on top 2-1.
Asked to sum up how he felt beating our Mersey rivals, Carragher said: "It's relief, really. Losing to Everton would have been an absolute nightmare.
"I was involved in their goal to start with and if we'd lost that 1-0 I don't think I'd ever have recovered, to be honest.
"We showed what Liverpool are about. David Moyes said the whole country was with them but we're a special club with special fans. Not many teams could come back.
"We did it before in 1986 when Everton were 1-0 up at half-time and we've done it again. We've got that ability to come back.
"Liverpool are expected to be in finals and win trophies. We've won one, let's win another."