The duo were speaking at the club's end of the season awards, and they looked to reassure fans ahead of Moyes's move to Manchester United this summer.
"When he came to me when we first met, he said chairman you are not going down, and he has kept us there and he has taken us forward," said Kenwright.
"Although we will miss like there is no tomorrow, I promise you David, we will just maintain what you have started and I won't let you down."
Moyes added: "I have always wanted to give the players all we can and I told them we would be at the top of the league and keep getting better.
"This group of players is great - they police themselves, in training every day they demand it and they will not allow that to change, they know the standards and they know what is required."
Everton Player of the Year Leighton Baines hopes the club can keep moving forward next season.
"We have performed well against the top sides and we have taken steps forward and have just fallen a little short of what we hoped."