The decision was expected, with the Goodison Park club keen to re-establish stability by promoting Elstone, who had been Wyness' deputy.
Owner and chairman Bill Kenwright has made the appointment following discussions with club officials in London, and after days of damaging speculation about the futures of both Kenwright and boss David Moyes.
Kenwright said: "Robert has been with us for more than three years and is, I believe, now fully conversant with the very special demands associated with running a club of our size and stature."