The Scot, who took over from Sir Alex Ferguson before the start of this season, has led the current champions to their worst start to a Premier League campaign.
Coleman, who joined Everton during Moyes's spell at Goodison Park, told Sky Sports News: "He got the job deservedly, he was recommended by Sir Alex himself, so I think he will definitely turn it around. It's only early days, only a few games into the season, so I am sure he will.
"It's been going quite well. [New Everton boss Roberto Martinez] has come in and he has put his own philosophy on the team, and we have done quite well to adapt to it.
"It's been a good start and we have all enjoyed it. It's different. You can see that he likes to play out from the back a bit more, but he has been great for us and hopefully we can have a good, successful season under him."
United sit ninth in the table, while Everton are seventh.