The 23-year-old became the most expensive British player in history with his £35million move from Newcastle in January 2011.
At the time shock was expressed at the size of the sum paid by Liverpool, who are widely accepted to have massively overpaid for the Englishman, but Carragher said that was not the player's fault.
Carroll scored the goal which put Liverpool into their second final of the season by heading home four minutes from time in the FA Cup semi-final with Everton at Wembley on Saturday.
He therefore maintained the record of Liverpool winning every time he has scored, albeit only in nine matches.
"It was a massive goal for Andy," Carragher said. "The fee was nothing to do with him and he can do nothing about that, but he is a young lad finding his feet at Liverpool.
"You forget how young the lad is and before he came to us he had only played six months in the Premier League."
Carragher was at fault for the opening goal as hesitation between him and Daniel Agger saw the Bootle-born defender hack a clearance against Tim Cahill and Nikica Jelavic convert the rebound.
So he was relieved to see the Reds come from behind, as they have done in the past on big occasions, with Luis Suarez equalising in the second half before Carroll snatched the winner.
"I was probably close to tears. Had we lost that game 1-0 I probably wouldn't have recovered," Carragher said. "Supporting Everton as a kid it was the biggest derby I've played in so I am delighted.
"I think that is in Liverpool's DNA (coming back from behind). I said to Stevie (Gerrard) 'How many times have we done that?'"
Gerrard also sprang to the defence of Carroll after the game, insisting criticism of the 23-year-old was unjustified.
"Andy has proved on the big stage what he is capable of and he has proved all his critics wrong," the England midfielder said.
"He was superb for us in the semi-final and has been every time he has been called upon. I am sure he will deliver for us in the final if he is called upon again."
Questions have been raised about the suitability of pairing Carroll and Suarez together, but Gerrard said the players had no doubts.
"Suarez and Carroll are two good players and they put in two good performances and showed what they are about," he said. "We have every faith in them as a partnership."