The Everton boss arrived at United's Carrington training ground on Monday for a meeting with Sir Alex Ferguson and the players as he began the process of familiarising himself with his new surroundings.
Moyes has pre-season to sort out his backroom team although it is not anticipated he is going to make any major changes.
Carrick revealed that the players are right behind the new man and are looking forward to the transition.
"As players we are embracing the challenge," said the United midfielder.
"For so long we have had the stability of knowing what to expect. We could turn up for pre-season and know how we would be preparing.
"There are going to be certain things that are new. But many will stay the same of course because there is a structure there and there is not a lot wrong.
"We are all looking forward to it. We are looking forward to the new manager coming in, working with him, improving as a squad and taking the club forward."
Carrick also confirmed that Ferguson has approached the fortnight since his retirement was confirmed in the same unemotional manner in private as he has in public.
"It has been business as usual," he said.
"He said a few thank yous after the game but generally it is as everyone has seen.
"He will have mixed emotions because it is a strange time for him, as it is for everyone at the club."