Moyes, who will leave Everton after 11 years at the end of the current season, has signed a six-year contract with the Barclays Premier League champions. The Scot will join United on 1 July.
Ferguson said: "When we discussed the candidates that we felt had the right attributes we unanimously agreed on David Moyes. 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. He was a young man then at the start of his career and has since gone on to do a magnificent job at Everton. There is no question he has all the qualities we expect of a manager at this club.”
“It’s a great honour to be asked to be the next Manager of Manchester United. 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.
“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.”
Everton confirmed earlier today Moyes had expressed a "desire to join Manchester United".
Before that, United's official press office account on Twitter confirmed Moyes would replace Ferguson via a Facebook page, before they deleted the tweet and took down the page.
During his tenure at Goodison Park, Moyes has taken charge of the Toffees on over 500 occasions, including four European campaigns.
The 50-year-old, hailed as an "outstanding manager", will manage Everton for the last time at Chelsea a week on Sunday.
Ferguson's final match in the dugout as United boss away to West Bromwich Albion will be his 1,500th game in charge of the Red Devils.
Moyes was actually considered by Ferguson to replace Brian Kidd as his assistant manager in 1999 while Preston North End manager.
Reflecting on Ferguson's interest 14 years ago, Moyes said last year: "When I went to speak to him about becoming his assistant years ago he thought me a little too intense."
When Ferguson was appointed United manager in November 1986, Moyes was playing for Bristol City.
United have won 13 titles, five FA Cups, four League Cups, two European Cups, a European Cup Winners' Cup, an Intercontinental Cup, a Club World Cup, a European Super Cup and 10 Community Shields.