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.
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"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.
"We couldn't stand in his way because he’s out of contract and it was his decision and he has made it.
"Its been an extraordinary 24 hours and it is important now that we finish off the season against West Ham and Chelsea, maintain our position and that from here on Everton Football Club continues, in the David Moyes and Everton Football Club tradition, with an equally significant manager."