Canaries boss Peter Grant told the club's official website: "I'm very disappointed to be losing Robert.
"We are trying to do what we can to maintain a promotion challenge and Robert was very much part of that.
"We were not informed of Derby's interest until late on Wednesday evening and the next day we received an offer at exactly the level of Robert's buy-out clause, which is disappointing and somewhat ironic."
Chief executive Neil Doncaster echoed Grant's disappointment at the sale of Earnshaw - who holds a unique record as the only player to score hat-tricks in all four divisions of the Football League and Premier League, the FA Cup, League Cup and at international level - but vowed that money from the sale would be handed to the Canaries boss.
Doncaster said: "We are sorry to be losing a player of Robert's ability but the search for other players goes on as we try to maintain a promotion challenge for this coming season.
"All of the Earnshaw money will go back to Peter Grant.
"The precise terms of that clause were critical to Robert's arrival here in January 2006 and without the buy-out clause he would not have come here."