Everton chairman Bill Kenwright and outgoing boss David Moyes have said the club will continue to progress despite the manager's departure to Manchester United.
Moyes is preparing for his final game in charge of the Toffees after 11 years at the helm - away to Chelsea on Sunday - before succeeding Sir Alex Ferguson at Old Trafford.
Speaking at Everton's end of season awards, Kenwright said: "When he came to me when we first met, he said: 'Chairman, you are not going down' - and he has kept us there [in the Premier League] and he has taken us forward.
"Although we will miss him like there is no tomorrow, I promise you, David, that we will just maintain what you have started and I won't let you down."
Moyes said his players "know the standards and know what is required", adding: "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."
Left-back Leighton Baines, named Player of the Year, said Everton had "taken steps forward and have just fallen a little short of what we hoped" this season.