The Scot had overseen the Premier League champions' worst start to a campaign in 24 years before a double from Adnan Januzaj on his top-flight debut handed United a 2-1 win over Sunderland.
But Coleman, who was handed his bow in English football by Moyes, is confident his ex-boss has the ability required to improve United's on-pitch fortunes.
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"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."
Following Moyes' departure after 11 years in charge at Goodison Park, Everton have thrived under new manager Roberto Martinez - with Saturday's 3-1 loss to Manchester City marking their first in the Premier League this season.
Coleman has hailed the Spaniard's impact at the club, and is enjoying the increased freedom allowed under Martinez's tactical approach.
The defender said: "It's been going quite well. He has come in and he has put his own philosophy on the team, and we have done quite well to adapt to it.
"We have got very good players. The [Manchester] City game [a 3-1 defeat on Saturday] was a disappointment, but hopefully we can put that right with the next game at home to Hull.
"But 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."