Moyes, 50, was recommended for the top job by Sir Alex and will quit his current club Everton at the end of the season to make the move.
"When we discussed the candidates that we felt had the right attributes, we unanimously agreed on David Moyes," Sir Alex said.
"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.
Sir Alex Ferguson
Sir Alex, 71, announced on Wednesday that he was retiring after 26 years in charge of Man United.
It had already been reported by talkSPORT that David Moyes had been chosen as his successor, ahead of other candidates like former Chelsea boss José Mourinho.
Moyes - another Scotsman, whose mother is from Portrush in Co Antrim - has led Everton on over 500 occasions since joining them from Preston North End in 2002.
His cousin Dessie Brown, secretary of Coleraine Football Club, told UTV: "David is very down-to-earth - he's a very level-headed man and he will have weighed up all the option.
"For him to even be considered at this stage, he himself will think it's a fantastic achievement."
He added: "The man just loves and eats and breathes and loves football."
Everton have already started the hunt for Moyes' replacement.
"The chairman, on behalf of the club, would like to place on record his thanks to David for the massive contribution he has made to Everton since his arrival in March 2002," a statement said.
"He has been an outstanding manager."
Moyes will be with Everton until the end of the season, with his final games in charge coming against West Ham and Chelsea, while Sir Alex will leave Manchester United after his 1,500th game in charge.
In David Moyes, we have someone who understands the things that make this such a special club.
Sir Bobby Charlton
In announcing his retirement, Sir Alex insisted: "The decision to retire is one that I have thought a great deal about and one that I have not taken lightly.
"It is the right time."
Speaking about the appointment of Moyes, Man United legend Sir Bobby Charlton said: "I have always said that we wanted the next manager to be a genuine Manchester United man.
"We have secured a man who is committed to the long-term and will build teams for the future as well as now. Stability breeds success."