The striker sealed a move to Upton Park from Liverpool after spending last season on loan at the club.
Cole expressed his delight at the move, suggesting that it is the kind of signing that shows the ambition of the club under manager Sam Allardyce.
"It is a boost for the club and shows where we are going," Cole told West Ham's official website.
"Andy Carroll is a young, England centre-forward and clubs do not usually sell them. He's a big talent.
"He did a good job and we just want him to get fit and come back because he is a big part of our team."
The Hammers paid £15m for Carroll, who has signed a six-year deal with the club.