Moyes, who was out of contract at Everton at the end of the season, has been handed a six-year contract by United.
"We unanimously agreed on David Moyes," said Ferguson.
"David is a man of great integrity with a strong work ethic. I've admired his work for a long time and approached him as far back as 1998 to discuss the position of assistant manager here.
"There is no question he has all the qualities we expect of a manager at this club."
Moyes said: "I am delighted that Sir Alex saw fit to recommend me for the job. I have great respect for everything he has done and for the football club.
"I know how hard it will be to follow the best manager ever, but the opportunity to manage Manchester United isn't something that comes around very often and I'm really looking forward to taking up the post next season."
He added: "I have had a terrific job at Everton, with a tremendous chairman and board of directors, and a great set of players. Between now and the end of the season, I will do everything in my power to make sure we finish as high as possible in the table.
"Everton's fantastic fans have played a big part in making my years at Goodison so enjoyable and I thank them wholeheartedly for the support they have given me and the players. Everton will be close to me for the rest of my life."