The defender has penned a two-and-a-half year deal at Goodison Park.
League Two Carlisle have said Branthwaite's move to Merseyside is for “a substantial fee".
In a statement director of football David Holdsworth said: “Ideally this fantastic young man would still be our player, but I think everyone will appreciate and understand that it can become very difficult for the individual when clubs at the highest level – some of the biggest clubs in the country – come calling and showing an interest in your services.
“We did what we could, but we have to respect the wishes of the player, his family and those who advise him.
"Once it becomes apparent that a player does want to move, all you can then do is work hard to get the best possible deal for the club.
“I'm sure we all wish Jarrad the very best as he makes this move to Everton and the Premier League."