The 24-year-old England international arrived at Upton Park from Liverpool last week in a deal thought to be worth around £15m.
"We saw some real quality glimpses from Andy last season and we can now look forward to him starting the season as a permanent West Ham player," Cottee told the Hammers' official website. "It's a big signing for the club.
"It's a massive statement of intent from the board and they need congratulating for bringing a player of Andy's standing to the club."
Carroll has signed a six-year deal with West Ham.