Chris Woods, who had been Everton's goalkeeping coach since 1998, takes on the same role with United and veteran coach Jimmy Lumsden also joins the staff after working with Moyes at Preston and Everton.
Moyes told United's official website: "I have worked with Steve, Chris and Jimmy for a number of years and I am delighted they have decided to join me at this great club.
"They bring great qualities in their respective fields and I know that, like me, they feel that this is a challenge to relish. I have great faith that together, we can build upon the success this club has enjoyed over many years."
Executive vice chairman Ed Woodward said: "I believe with David and his new team in place, along with our current backroom staff, we can now start preparing for the new season ahead.
"This is an exciting and challenging time for the club and I have great belief that David and his team will have all the energy and attributes needed to help this club continue in its long-standing traditions."