The Toffees will have a new man in charge for the first time in 11 years next season after Moyes accepted the opportunity to succeed Sir Alex Ferguson at Manchester United.
Moyes is widely regarded to have done an excellent job at Goodison Park despite failing to win a trophy, and goalkeeper Howard sees no reason it cannot be a smooth transition.
Howard told the Liverpool Post: "We are a really strong group of quality players and the foundations are set.
"The table is set for the next guy and I have no doubt, if he is a strong character, a guy who can lead us well, we will have no problems.
"Manchester United and Everton are in the same boat with having to replace the manager in terms of expectation and all the rest.
"It's just very rare to replace one long-serving manager with another long-serving one so we will need to fill the vacancy with a very strong character. And I have no doubt that will happen."
Howard revealed there was little ceremony when Moyes, who will take charge of his final Everton game at Chelsea on Sunday, told his players last week that he would be leaving.
"When he told us, it was quick and simple," said the goalkeeper. "He does not do anything dramatic, just straight to the point, and then it's 'let's go training'.
"We already knew by that time so we were just happy for him really. What an opportunity. I have no doubt he will make the most of it and we are just trying to focus on the last games and then see what our own future holds."