The Toffees chief is determined to find the right man to succeed the Scot, who will replace the retiring Sir Alex Ferguson at Manchester United at the end of the season
Moyes will replace the retiring Sir Alex Ferguson at Old Trafford, who announced his retirement on Wednesday, but focus at Goodison Park now switches to a managerial hunt of their own.
Moyes' exit ends his 11-year tenure at the club, and Kenwright believes the 50-year-old has left a lasting legacy and the daunting task to find a worthy successor is already underway.
He told Sky Sports News: "We really will be out there finding the right man for this football club because it’s an extraordinary football club that is worshipped by its fans. They are the people – more than anything - that I have been thinking of over the last 24 hours.
"Eleven years ago I made a decision and it was an instant decision when I met David. I don’t think that can happen this time. We have got to be out there looking to see what kind of candidates can take the club forward.
"We've had a really good 11 year adventure with David and now it is my job and the club's job to get a worthy successor and build on what David has brought to the club and to hopefully move on from there."
Although Moyes did not win a trophy during his time on Merseyside, the former Preston manager won plaudits for working wonders on a shoestring budget and constantly over-achieving with his side.
Kenwright was full of praise for a manager that took charge of over 500 matches and believes his new employers are inheriting the perfect replacement for Ferguson.
"He's a great manager and Manchester United are very lucky," added the chairman.
"I'm pretty certain Evertonians will only look on David with gratitude. We have got two games left and it will be tough for those Evertonians to say goodbye to him."