Saban is expected to be formally introduced at a press conference in Miami on Monday.
"We're pleased to have Nick Saban as the next head coach of the Miami Dolphins, and we're looking forward to introducing him to our fans," team owner Wayne Huizenga said in a prepared statement released by the Dolphins.
"I also want to thank two outstanding individuals, Jim Bates, for his remarkable leadership of our football team during a very difficult time, and Art Shell, who had an extremely impressive interview."
Both Bates and Shell had interviews with Huizenga and team president Eddie Jones this week, but it was clear all along that the guy the Dolphins wanted was Nick Saban.
His contract will be a five-year deal and is believed to be worth between $4.5 and $5 million a year. Saban will have full authority over personnel matters, and he's expected to bring in former Tennessee exec Floyd Reese as a new GM or team president.
Saban said he would address the issue of the coaches on the current staff at a later time.
But the big step was taken on Saturday when the Dolphins, even though it took a while, got their man.