The Scot has found himself under pressure just weeks after replacing Sir Alex Ferguson 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."