Peters made departing Everton boss Moyes his assistant-manager at Preston in 1994 and stepped aside for the Scotsman to take full control four years later.
"When I took over as manager at Preston, I made David my assistant straight away, even though he was still playing," Peters told the Mirror. "He had all the qualities you would want.
"He was solid, trustworthy, honest and, most of all, hardworking. From day one, he was like a young Sir Alex Ferguson because everything was about hard work and setting good examples.
"I had a young squad at Preston, David was champing at the bit to be a manager and I just felt it was right to stand aside and let him do it. David was like a younger, less experienced version of Sir Alex straight away. I witnessed it because I stayed at Preston.
"David ran the club from top to bottom. The way he managed people and different personalities was similar to Sir Alex and he had the same attention to every single tiny detail.
"He will uphold all the values and good working practices.
"David has many, many qualities but his best as a manager is that when he closes the dressing-room door everything stays in there.
"The United players have been used to that under Sir Alex and it will not change under David."