The Scot has found himself under pressure just weeks into his tenure at Old Trafford and needed a double from teenager Adnan Januzaj at lowly Sunderland on Saturday to prevent his side slipping to a third successive Premier League defeat for the first time since December 2001.
However Coleman, who was given his big chance by Moyes at Everton, is confident he will make a success of his new job.
The Republic of Ireland full-back, who will hope to celebrate his 25th birthday on Friday by taking on the might of Germany in a World Cup qualifier, said: "Just from my time with him, all I can say is he has been a top manager and he has done so well with Everton.
"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."
Ironically, the club Moyes left behind currently sits in seventh position in the Premier League table, two places and as many points better off than the reigning champions, with his replacement Roberto Martinez having made a flying start to life on the blue half of Merseyside.
The Spaniard's recruitment of Gareth Barry, James McCarthy and Romelu Lukaku and his methods have given the Toffees a different outlook, and Coleman and his team-mates have responded in impressive fashion.
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."
It has been all change on the international, as well as club, front for Coleman in recent months with Giovanni Trapattoni's reign as Ireland manager having come to an end after last month's qualifier defeats by Sweden and Austria.
The Republic can effectively no longer make it to Brazil next summer and head into Friday's game in Cologne and Tuesday's home clash with Kazakhstan with no permanent successor yet appointed.
Under-21s boss Noel King, for whom Coleman made one appearance before his elevation to the senior squad, will take charge of the two games, but the Everton man admits he was sad to see Trapattoni go.
He said: "He was great with me. He gave me my first cap and he gave me plenty after that, so I never had any problems with him.
"Obviously, it's sad to see a manager go, but we have to look forward now and the players have to impress on Friday night and Tuesday night because the new manager could be watching."
However for Coleman, one of the few survivors of last October's 6-1 mauling by the Germans at the Aviva Stadium, there is a wrong to be righted as well as a potential new manager to impress.
He said: "Obviously we want to go in and make sure that doesn't happen again, but we won't go in with the hangover from it.
"We will go in hoping to impress the new manager who's watching, and obviously to do a good job for Noel as well on a big night for him.
"It's his first time in charge of the senior national team, so we want to do as best we can for him as well."