As exclusively revealed by Goal following the announcement of the 71-year-old's retirement, the Toffees boss had been identified as the outgoing manager's successor.
Moyes has agreed a six-year deal at the club, which will start on June 1, and will take his place at the Old Trafford helm with the blessing of Sir Alex.
"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."
Everton announced earlier on Thursday that Moyes was set to leave Goodison Park at the end of the season.
The Scot acknowledged that Sir Alex will be a tough act to follow after accruing 38 trophies during his 26-year reign, but admitted that the opportunity could not be passed up.
Moyes added: "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 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."
Moyes is now set to leave Everton after over a decade in charge, and the 50-year-old has thanked the club for their support and guidance during his tenure.
"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."
United chief executive, David Gill, added his own congratulations and backed Moyes to become a success at Old Trafford.
"I'm delighted that David has agreed to join Manchester United. He has done a terrific job at Everton and has all the qualities that we are looking for in a manager to succeed Alex."
Goal.com exclusively revealed on Wednesday that Wigan manager Roberto Martinez is Everton's No.1 target to replace the outgoing Moyes.